A word of caution. Of course at the final step $\forall x\neg\neg Ax$ entails (or at least, classically entails) $\forall x Ax$, and does so because adjacent double negations (classically) cancel each other out.
The inference is not strictly an application of a standard double negation rule of the form from $\neg\neg\varphi$ infer $\varphi$. That rule only allows us to remove initial double negations.
To show the entailment in standard proof systems requires a three-step mini-proof:
$\forall x\neg\neg Ax\quad$ (given)
$ \neg\neg Aa\quad\quad$ (universal instantiation with parameter or free variable depending on system)
$ Aa\quad\quad\quad$ (NOW you can apply the DN rule)
$\forall x Ax\quad\quad$ (universal generalization)
So your reasoning is informally just fine, but do be careful about jumping from $\forall x\neg\neg Ax$ to $\forall x Ax$ in formal proofs!