There are examples in arbitrary dimension. Begin with the polynomial ring $S=k[N]=k[s_1,\dots,s_d]$, i.e., the semigroup $k$-algebra on the semigroup $N=(\mathbb{Z}_{\geq 0})^d$ of exponent vectors for monomials in $S$. Now for any subsemigroup $M\subset N$, let $R=k[M]\subset k[N]$ be the corresponding $k$-subalgebra of $S$. These give plenty of examples.

For instance, let $R$ be the $k$-subalgebra generated by monomials of total degree $\geq d+1$. Let $\mathfrak{m}$ be the maximal ideal generated by all monomials of total degree $\geq d+1$. Let $I$ be the principal ideal generated by $x=s_1^2s_2\cdots s_d$. Then the element $y=s_1^3s_2\cdots s_d$ in $R$ is in $q(I)$, but it is not in $I$. Indeed, $y^{d+1}$ equals $x^{d+1}\cdot s_1^{d+1}$, so $y$ is in the integral closure of $I$. Also, for every monomial $m$ of degree $\geq d+1$, then $my$ equals $(s_1m)x$, where $s_1m$ is another monomial in $R$. Thus $y$ is also in the saturation of $I$ with respect to the maximal ideal $\mathfrak{m}$. Please note also: $R$ is regular in codimension $\leq d-1$, but it is definitely not $S2$.

There are some exercises in Eisenbud's "Commutative Algebra" that discuss these types of rings and their integral closures.