Here is a another approach.
For convenience I write $n$ instead of $N$, and $A_n$ for $A$.

By definition

$$\det(A_n) = \sum_{\pi\in S_n} \operatorname{sign}(\pi) \prod_{i=1}^n a_{i,\pi(i)}$$

By assumption $A_n$ is symmetric and the $a_{i,j}$ are (for $j\geq i$) mutually independent random variables $X_{i,j}$
with $\mathbb{E}(X_{i,i})=p=:a$, $\mathbb{E}(X_{i,j})=p^2=:c$, and $\mathbb{E}(X_{i,j}^2)=p^4 +\frac{p^2(1-p^2)}{n}=:b$ for $j\not =i$.

Denote by $C(\pi)$ the set of cycles of $\pi$ and write
$\prod_{i=1}^n X_{i,\pi(i)}= \prod_{C \in C(\pi)} X_C$, with $X_C:=\prod_{i\in C} X_{i,\pi(i)}$. The cycles are disjoint, thus
the different $X_C$ are independent, further
$$\mathbb{E}(X_C)=\left\{ \begin{array}{cr} a &\mbox{ if $C$ is a fixed point}\\
b &\mbox{ if $C$ is a 2-cycle (transposition)}\\
c^{|C|} &\mbox{ if the length $|C|$ of $C$ is $\geq 3$}\end{array}\right.$$

To use that, denote by $Z_i(\pi), \,1\leq i\leq n$ the no. of cycles of length $i$ of $\pi$ and by $Z(\pi)=\sum_{i=1}^n Z_i(\pi)$
denote the total no. of cycles of $\pi$, and recall that $\operatorname{sign}(\pi)=(-1)^{n-Z(\pi)}$ ( "a permutation $\pi\in S_n$ is even iff
it's decrement $n-Z(\pi)$ is even"). We then have

$$\mathbb{E}\det(A_n)=\sum_{\pi\in S_n} (-1)^{n-Z(\pi)} a^{Z_1(\pi)} b^{Z_2(\pi)} c^{n-Z_1(\pi)-2Z_2(\pi)}$$

It is well known how to handle this type of sums. Standard generating function techniques (see e.g. section 5.2 in Stanley's EC 2) show that
$$\sum_{\pi \in S_n} t^{Z(\pi)} t_1^{Z_1(\pi)}\ldots{t_n}^{Z_n(\pi)}=n! [z^n] e^{\sum_{i=1}^n t\,t_i\frac{z^i}{i}}$$
Using that and simple manipulations we find
$$\mathbb{E}\det(A_n)=n!\,[z^n] (1+cz)\,e^{(a-c)z-(b-c^2)\frac{z^2}{2}}$$
The Hermite polynomials $He_n(x)$ are defined by $He_n(x)=n![t^n] e^{xt-\frac{t^2}{2}}$. Now note that with $v:=b-c^2$ (the variance of the off-diagonal matrix elements)
and $d:=a-c$ (the difference of the expectations) you have
$$\mathbb{E}\det(A_n)=\sqrt{v^n}\,He_n(\frac{d}{\sqrt{v}})+cn\sqrt{v^{n-1}}\,He_{n-1}(\frac{d}{\sqrt{v}})$$

The asymptotic properties of the Hermite polynomials have been intensively investigated, so you can probably find precise results for your concrete setting in the literature.

EDIT: Note that this doesn't answer your question for the expectation of the **absolute value** of the determinant (thanks to Carlo Beenakker for pointing this out). I'll still leave it here in case someone's interested in the expectation of the determinant.