Now let $v$ be $s_ny$ in $C$. Let $w$ be $v-s_1$. Thus, the $A$-subalgebra $A[w]$ of $C$ equals the $A$-subalgebra $A[v]$. The element $v$ satisfies the monic polynomial equation $q_n(v)=0$, $$q_n(v) = v^n-s_1s_n^1v^{n-1} + \dots + (-1)^{n-1}s_{n-1}s_n^{n-1}v + (-1)^ns_n^n.$$ By direct computation, for a variable $y$, the ring $B=A[y]/\langle q_n(y) \rangle$ is a complete intersection ring that a finite, flat extension of $A$, and whose singular locus is an irreducible closed subset of the spectrum with associated prime ideal $\langle s_n,v\rangle$. Since $A\to B$ is finite and flat, every irreducible component of $\text{Spec}(B)$ dominates $\text{Spec}(A)$. However, after inverting $s_n$, the natural map $B[1/s_n]\to C[1/s_n]$ is an isomorphism. Hence, $B$ is generically reduced and has a single irreducible component. Since $B$ is a complete intersection ring, it is everywhere reduced. Thus $B$ is an integral domain, hence embeds in its fraction field. Therefore, since the map of fraction rings, $B[1/s_n]\to C[1/s_n]$, is injective, also $B\to C$ is injective. In conclusion, the natural map $B\to A[v]$ is both injective and surjective, hence an isomorphism.

The quotient ring $C/\langle w \rangle$ is $$k[s_1,\dots,s_{n-1},x]/\langle r_n \rangle, \ \ r_n = (1-x^n)s_1 + x^{n-1}s_2 - x^{n-2}s_3 + \dots + (-1)^{n}x^2s_{n-1} + x^{n+1}. $$ By the Jacobian criterion, this ring is smooth over $k$. Moreover, the ring extension $k[x]\to C/\langle w \rangle$ is an affine space bundle (over $D(x)$, resp. $D(1-x^n)$, it is trivialized). Thus, it is irreducible of dimension $n-1$. Since the quotient ring is regular of dimension $n-1$, the ideal $wC$ in $C$ is a prime ideal. Therefore $w$ is a prime element of $C$.

The argument that $B/\langle w \rangle$ is integral of dimension $n-1$ is similar, but slightly more involved. The quotient ring is $$B/\langle w \rangle = k[s_1,\dots,s_{n-1}]/\langle g_n \rangle,$$ $$g_n = (1-s_n)s_1^n + s_1^{n-2}s_ns_2 + \dots + (-1)^{n-1}s_1s_n^{n-1}s_{n-1} + (-1)^ns_n^n.$$ By the Jacobian criterion, the singular locus comes from the prime ideal $\langle s_1,s_n \rangle$. So the ring is reduced. Moreover, the ring extension $k[s_1,s_{n-1}] \to B/\langle w \rangle$ is flat of relative dimension $n-3$ away from $\langle s_1, s_{n-1}\range$, with the fiber over $\langle s_1,s_{n-1}\rangle$ of dimension $n-2$. Since a closed subset of dimension $n-2$ cannot be an irreducible component of the complete intersection ring $B/\langle w \rangle$ of dimension $n-1$, it follows that every fiber dominates $\text{Spec}k[s_1,s_{n-1}]\setminus \langle s_1,\s_{n-1} \rangle$. The restriction over this open set is again an affine space bundle, hence it is irreducible. Therefore, since $B/\langle w \rangle$ is reduced and irreducible, the element $w$ of $B$ is prime.

Now let $v$ be $s_nx$ in $C$. Let $w$ be $v-s_1$. Thus, the $A$-subalgebra $A[w]$ of $C$ equals the $A$-subalgebra $A[v]$. The element $v$ satisfies the monic polynomial equation $q_n(v)=0$, $$q_n(v) = v^n-s_1s_n^1v^{n-1} + \dots + (-1)^{n-1}s_{n-1}s_n^{n-1}v + (-1)^ns_n^n.$$ By direct computation, for a variable $y$, the ring $B=A[y]/\langle q_n(y) \rangle$ is a complete intersection ring that a finite, flat extension of $A$, and whose singular locus is an irreducible closed subset of the spectrum with associated prime ideal $\langle s_n,v\rangle$. Since $A\to B$ is finite and flat, every irreducible component of $\text{Spec}(B)$ dominates $\text{Spec}(A)$. However, after inverting $s_n$, the natural map $B[1/s_n]\to C[1/s_n]$ is an isomorphism. Hence, $B$ is generically reduced and has a single irreducible component. Since $B$ is a complete intersection ring, it is everywhere reduced. Thus $B$ is an integral domain, hence embeds in its fraction field. Therefore, since the map of fraction rings, $B[1/s_n]\to C[1/s_n]$, is injective, also $B\to C$ is injective. In conclusion, the natural map $B\to A[v]$ is both injective and surjective, hence an isomorphism.

The quotient ring $C/\langle w \rangle$ is $$k[s_1,\dots,s_{n-1},x]/\langle r_n \rangle, $$ $$r_n = (1-x^n)s_1 + x^{n-1}s_2 - x^{n-2}s_3 + \dots + (-1)^{n}x^2s_{n-1} + x^{n+1}. $$ By the Jacobian criterion, this ring is smooth over $k$. Moreover, the ring extension $k[x]\to C/\langle w \rangle$ is an affine space bundle (over $D(x)$, resp. $D(1-x^n)$, it is trivialized). Thus, it is irreducible of dimension $n-1$. Since the quotient ring is regular of dimension $n-1$, the ideal $wC$ in $C$ is a prime ideal. Therefore $w$ is a prime element of $C$.

The argument that $B/\langle w \rangle$ is integral of dimension $n-1$ is similar, but slightly more involved. The quotient ring is $$B/\langle w \rangle = k[s_1,\dots,s_{n-1}]/\langle g_n \rangle,$$ $$g_n = (1-s_n)s_1^n + s_1^{n-2}s_ns_2 + \dots + (-1)^{n-1}s_1s_n^{n-1}s_{n-1} + (-1)^ns_n^n.$$ By the Jacobian criterion, the singular locus comes from the prime ideal $\langle s_1,s_n \rangle$. So the ring is reduced. Moreover, the ring extension $k[s_1,s_{n-1}] \to B/\langle w \rangle$ is flat of relative dimension $n-3$ away from $\langle s_1, s_{n-1}\rangle$, with the fiber over $\langle s_1,s_{n-1}\rangle$ of dimension $n-2$. Since a closed subset of dimension $n-2$ cannot be an irreducible component of the complete intersection ring $B/\langle w \rangle$ of dimension $n-1$, it follows that every fiber dominates $\text{Spec}k[s_1,s_{n-1}]\setminus \langle s_1,s_{n-1} \rangle$. The restriction over this open set is again an affine space bundle, hence it is irreducible. Therefore, since $B/\langle w \rangle$ is reduced and irreducible, the element $w$ of $B$ is prime.

Edit. The OP asks for a different example. In the following example, both $A\to A[w]$ and $A\to C$ are finite and flat. Both $A$ and $C$ are regular Noetherian integral domains. Yet $A[w]$ is not even integrally closed in its fraction field, because it fails the condition $R1$ of Serre's criterion. That is the common feature of both examples: a very singular scheme can be "sandwiched" between two smooth schemes. Once you understand the geometric nature of these examples, you can produce many other examples for yourself.

Let $n\geq 5$ be an integer that is prime to the characteristic. Let $A$ be $k[s_1,\dots,s_n]$, a polynomial ring in $n$ variables. Define $C$ to be the following $A$-algebra, $$C = A[x]/\langle p_n(x) \rangle, \ \ p_n(x) = x^n - s_1x^{n-1} + \dots + (-1)^{n-m}s_mx^{n-m} + \dots + (-1)^ns_n.$$ The equation $p_n(x)$ allows us to "eliminate" $s_n$, so that $C$ is isomorphic to $k[s_1,\dots,s_{n-1},x]$, a polynomial ring in $n$ variables. Thus, both $A$ and $C$ are Noetherian, regular, integral domains of dimension $n$. By construction, $C$ is a finite flat extension of $A$ of rank $n$ as an $A$-module.

Now let $v$ be $s_ny$ in $C$. Let $w$ be $v-s_1$. Thus, the $A$-subalgebra $A[w]$ of $C$ equals the $A$-subalgebra $A[v]$. The element $v$ satisfies the monic polynomial equation $q_n(v)=0$, $$q_n(v) = v^n-s_1s_n^1v^{n-1} + \dots + (-1)^{n-1}s_{n-1}s_n^{n-1}v + (-1)^ns_n^n.$$ By direct computation, for a variable $y$, the ring $B=A[y]/\langle q_n(y) \rangle$ is a complete intersection ring that a finite, flat extension of $A$, and whose singular locus is an irreducible closed subset of the spectrum with associated prime ideal $\langle s_n,v\rangle$. Since $A\to B$ is finite and flat, every irreducible component of $\text{Spec}(B)$ dominates $\text{Spec}(A)$. However, after inverting $s_n$, the natural map $B[1/s_n]\to C[1/s_n]$ is an isomorphism. Hence, $B$ is generically reduced and has a single irreducible component. Since $B$ is a complete intersection ring, it is everywhere reduced. Thus $B$ is an integral domain, hence embeds in its fraction field. Therefore, since the map of fraction rings, $B[1/s_n]\to C[1/s_n]$, is injective, also $B\to C$ is injective. In conclusion, the natural map $B\to A[v]$ is both injective and surjective, hence an isomorphism.

Note that $B$ is not integrally closed in its fraction field. Of course we can find a specific fraction that satisfies a monic equation yet is not in $B$. However, it is faster to use Serre's Criterion. Since $B$ is a complete intersection ring, it is Cohen-Macaulay, and thus it is $S2$. Therefore $B$ is integrally closed in its fraction field if and only if $B$ is $R1$. However, the singular locus of $B$ is the irreducible closed subset with coordinate ring $$B/\langle v,s_n \rangle = A/\langle s_n \rangle \cong k[s_1,\dots,s_{n-1}].$$ This has dimension $n-1$. So $B$ is singular in codimension $1$, thus it is not $R1$. Since $B$ is not integrally closed in its fraction field, by Gauss's Lemma, $B$ is also not a unique factorization domain.

The fraction field extension $\text{Frac}(A)\to \text{Frac}(C)$ is a degree $n$ extension whose associated Galois group is the full symmetric group $\mathfrak{S}_n$. In particular, for $n\geq 5$, this cannot be a solvable extension, much less a radical extension. Thus, $\text{Frac}(A)$ is "radically closed" in $\text{Frac}(C)$. Since $A$ is integrally closed in its fraction field (by Gauss's Lemma again), also $A$ is "radically closed" in $\text{Frac}(C)$.

The quotient ring $C/\langle w \rangle$ is $$k[s_1,\dots,s_{n-1},x]/\langle r_n \rangle, \ \ r_n = (1-x^n)s_1 + x^{n-1}s_2 - x^{n-2}s_3 + \dots + (-1)^{n}x^2s_{n-1} + x^{n+1}. $$ By the Jacobian criterion, this ring is smooth over $k$. Moreover, the ring extension $k[x]\to C/\langle w \rangle$ is an affine space bundle (over $D(x)$, resp. $D(1-x^n)$, it is trivialized). Thus, it is irreducible of dimension $n-1$. Since the quotient ring is regular of dimension $n-1$, the ideal $wC$ in $C$ is a prime ideal. Therefore $w$ is a prime element of $C$.

The argument that $B/\langle w \rangle$ is integral of dimension $n-1$ is similar, but slightly more involved. The quotient ring is $$B/\langle w \rangle = k[s_1,\dots,s_{n-1}]/\langle g_n \rangle,$$ $$g_n = (1-s_n)s_1^n + s_1^{n-2}s_ns_2 + \dots + (-1)^{n-1}s_1s_n^{n-1}s_{n-1} + (-1)^ns_n^n.$$ By the Jacobian criterion, the singular locus comes from the prime ideal $\langle s_1,s_n \rangle$. So the ring is reduced. Moreover, the ring extension $k[s_1,s_{n-1}] \to B/\langle w \rangle$ is flat of relative dimension $n-3$ away from $\langle s_1, s_{n-1}\range$, with the fiber over $\langle s_1,s_{n-1}\rangle$ of dimension $n-2$. Since a closed subset of dimension $n-2$ cannot be an irreducible component of the complete intersection ring $B/\langle w \rangle$ of dimension $n-1$, it follows that every fiber dominates $\text{Spec}k[s_1,s_{n-1}]\setminus \langle s_1,\s_{n-1} \rangle$. The restriction over this open set is again an affine space bundle, hence it is irreducible. Therefore, since $B/\langle w \rangle$ is reduced and irreducible, the element $w$ of $B$ is prime.

I want to emphasize once more the geometric nature of the example. You have a number of hypotheses -- "radically closed", prime element in $B$, etc. These are "generically" satisfied. So if you write down a family of complete intersection rings that fail $R1$ yet are sandwiched between factorial rings, "typically" all of your hypotheses should hold. In the example above, both $A\to B$ and $A\to C$ are finite and flat. So neither finiteness nor flatness are the issue. The issue is that you can "pinch" $C$ to make a complete intersection ring that is not $R1$.