The conjectures are true. If you negate the first row of $A_p^{-}$ you get a cofactor of the matrix in "Chapman's evil determinant". In particular you can get the answer from the same matrix decomposition used by Vsemirnov in his paper: On the evaluation of R. Chapman’s “evil determinant”.
I couldn't deduce the result for $A_p^{+}$ from Vsemirnov's paper so I'm sketching a different approach here. The proof essentially consist of (1) using quadratic Gauss sums to write the matrix as a (rank one perturbation of) a product of shifted Vandermonde matrices, and (2) repeated use of the Lagrange interpolation formula to evaluate various sums of rational functions evaluated at roots of unity. Here are the details:
Throughout this post $p$ is a prime $3\pmod{4}$ and $\zeta=e^{\frac{2\pi i}{p}}$.
Lemma 1: Given $a_1,a_2,\dots,a_{\frac{p-1}{2}},b_1,b_2,\dots,b_{\frac{p-1}{2}}\in \mathbb Z/p\mathbb Z$ we have
$$\left[\left(\frac{a_i +b_j}p\right)\right]_{1\le i,j\le \frac{p-1}{2}}=\frac{1}{i\sqrt{p}}\left(J+2M_aM_b^{\top}\right)$$
where $M_a=[\zeta^{a_i k^2}]_{1\le i,k\le \frac{p-1}{2}}$ and $M_b=[\zeta^{b_jk^2}]_{1\le j,k\le \frac{p-1}{2}}$ and $J$ is the matrix with all entries $1$ which can also be written as $\textbf{1}^{\top}\textbf{1}$, with $\textbf{1}$ being the row vector of all $1$'s.
Proof: This follows entry-wise from the quadratic Gauss sum
$$\left(\frac{a_i +b_j}p\right)=\frac{1}{i\sqrt p}\left(1+2\sum_{k=1}^{\frac{p-1}{2}}\zeta^{(a_i+b_j)k^2}\right)=\frac{1}{i\sqrt p}\left(1+2\sum_{k=1}^{\frac{p-1}{2}}\zeta^{a_ik^2}\cdot \zeta^{k^2 b_j}\right).$$
Corollary 1: By the matrix determinant lemma, we have
$$\det \left[\left(\frac{a_i +b_j}p\right)\right]_{1\le i,j\le \frac{p-1}{2}}=\frac{1}{\left(i\sqrt p\right)^{\frac{p-1}{2}}}\left(2^{\frac{p-1}{2}}\det M_a \det M_b+\textbf{1}^{\top}\textbf{Adj} (2M_aM_b^{\top}) \textbf{1}\right)$$
$$=\frac{1}{\left(i\sqrt p\right)^{\frac{p-1}{2}}}\left(2^{\frac{p-1}{2}}\det M_a \det M_b+2^{\frac{p-3}{2}}\sum_{r=1}^{\frac{p-1}{2}}\det M_a^{(r)}\det M_b^{(r)}\right)$$
Where we use the notation $M^{(r)}$ to denote the matrix obtained from $M$ by replacing the $r$-th row by $\textbf{1}$.
The matrix $A_p^{+}$ corresponds to the choice $(a_1,\dots,a_{\frac{p-1}{2}})=(0,2,3,\dots,\frac{p-1}{2})$ and $(b_1,\dots,b_{\frac{p-1}{2}})=(1,2,\dots,\frac{p-1}{2})$. Computing $\det M_a,\det M_b$ or $\det M_a^{(r)},\det M_b^{(r)}$ for these vectors is an evaluation of Vandermonde determinants or simple instances of Schur polynomials, so I am leaving it as an exercise, for the sake of brevity. What you will end up getting from corollary 1 on these specific vectors, is the following:
$$\det A_p^{+}=$$
$$\frac{\left(\prod_{k_1<k_2} (\zeta^{k_1^2}-\zeta^{k_2^2})\right)^2}{\left(i\sqrt p\right)^{\frac{p-1}{2}}}\left(-2^{\frac{p-3}{2}} \left(i\sqrt{p}+1\right)+2^{\frac{p-3}{2}}\sum_{r=1}^{\frac{p-1}{2}}F_{+}(\zeta^{r^2})\left(\prod_{k\neq r} \frac{1-\zeta^{k^2}}{\zeta^{r^2}-\zeta^{k^2}}\right)^2\right)$$
where $F_{+}(\zeta^{r^2})=\frac{1-i\sqrt p}{2\zeta^{2r^2}}-\frac{1}{\zeta^{3r^2}}$. What follows are some facts that help us simplify this expression:
Lemma 2: We have $\left(\prod_{k_1<k_2}(\zeta^{k_1^2}-\zeta^{k_2^2})\right)^2=(i\sqrt p)^{\frac{p-3}{2}}$.
Proof: This calculation follows from the fact that for any $n\neq 0\pmod p$, the equation $n=x^2-y^2$ has exactly $\frac{p-3}{4}$ solutions with $x,y\in \{1,2,\dots,\frac{p-1}{2}\}$.
Lemma 3 We have the following equalities
$$\sum_{r=1}^{\frac{p-1}{2}}\frac{1}{\zeta^{2r^2}}\left(\prod_{k\neq r} \frac{1-\zeta^{k^2}}{\zeta^{r^2}-\zeta^{k^2}}\right)^2=-2$$
$$\sum_{r=1}^{\frac{p-1}{2}}\frac{1}{\zeta^{3r^2}}\left(\prod_{k\neq r} \frac{1-\zeta^{k^2}}{\zeta^{r^2}-\zeta^{k^2}}\right)^2=-2+i\sqrt{p}$$
Proof: For the first equality we will us Lagrange interpolation on the polynomial $G(x)=\frac{1}{x}\left(\prod_{k=1}^{\frac{p-1}{2}} (x-\zeta^{-k^2})-\prod_{k=1}^{\frac{p-1}{2}}(x-\zeta^{k^2})\right)$ at the points $x=\zeta^{1^2},\dots,\zeta^{(\frac{p-1}{2})^2}$. On one hand we have $G(1)=2\prod_{k=1}^{\frac{p-1}{2}}(1-\zeta^{-k^2})$, and on the other
$$G(1)=\sum_{r=1}^{\frac{p-1}{2}}\frac{1}{\zeta^{r^2}}\prod_{k=1}^{\frac{p-1}{2}}(\zeta^{r^2}-\zeta^{-k^2})\prod_{k\neq r} \frac{1-\zeta^{k^2}}{\zeta^{r^2}-\zeta^{k^2}}$$
$$=\sum_{r=1}^{\frac{p-1}{2}}\frac{1}{\zeta^{r^2}}\left(\frac{-p\zeta^{(p-1)r^2}}{\prod_{k=1}^{\frac{p-1}{2}}(1-\zeta^{k^2})}\right)\left(\prod_{k\neq r} \frac{1-\zeta^{k^2}}{\zeta^{r^2}-\zeta^{k^2}}\right)^2$$
So we have
$$\sum_{r=1}^{\frac{p-1}{2}}\frac{1}{\zeta^{2r^2}}\left(\prod_{k\neq r} \frac{1-\zeta^{k^2}}{\zeta^{r^2}-\zeta^{k^2}}\right)^2=\frac{-2}{p}\prod_{k=1}^{\frac{p-1}{2}}(1-\zeta^{-k^2})(1-\zeta^{k^2})=-2.$$
The second identity can be similarly obtained by applying Lagrange interpolation to $G(x)$ at the points $x=0,\zeta^{1^2},\zeta^{2^2},\dots,\zeta^{(\frac{p-1}{2})^2}$. For this you will need that
$$G(0)=\zeta^{-1^2-\cdots-(\frac{p-1}{2})^2}\left(\sum_{k=1}^{\frac{p-1}{2}}\zeta^{-k^2}\right)-\zeta^{1^2+\cdots+(\frac{p-1}{2})^2}\left(\sum_{k=1}^{\frac{p-1}{2}}\zeta^{k^2}\right)$$
$$=\frac{-1-i\sqrt{p}}{2}-\frac{-1+i\sqrt p}{2}=-i\sqrt{p}.$$
Applying the calculations from the previous two lemmas to our expression gives
$$\det A_p^{+}=\frac{2^{\frac{p-3}{2}}}{i\sqrt{p}}\left(-i\sqrt{p}-1-2\left(\frac{1-i\sqrt{p}}{2}\right)+2-i\sqrt{p}\right)=-2^{\frac{p-3}{2}}$$
as desired.
