Assuming consistency of $\text{ZFC}$ (with some large cardinal axiom), is the following statement consistent with $\text{ZFC}$?
Any $\Sigma_1$ statement with parameters $\omega_1,\omega_2$ which holds in a $\omega_1,\omega_2$ preserving forcing extension holds in $V$.
The answer is yes. Your theory holds in any model of the Maximality Principle for $(\omega_1,\omega_2)$-preserving forcing. So indeed, one can get a stronger result.
Specifically, this version of the maximality principle asserts that whenever a statement $\sigma$ is forceable by $(\omega_1,\omega_2)$-preserving forcing, in such a way that it remains true in all such further extensions, then it is already true in $V$. Thus, it can be expressed modally as the scheme of assertions
$$\Diamond\square\sigma\to\sigma,$$
where $\Diamond\psi$ means that $\psi$ is forceable by $(\omega_1,\omega_2)$-preserving forcing, and $\square\psi$ means that $\psi$ holds in all such forcing extensions. The main point here, in connection with your question, is that the forceable $\Sigma_1$ assertions are of course forceably necessary, since once you force them, they remain true in all further extensions. Thus, your theory is generalized by the theory asserting that all forceably necessary statements are already true.
Theorem. The maximality principle $\text{MP}(\omega_1,\omega_2\text{-preserving})$ is equiconsistent with $\text{ZFC}$.
Proof. Suppose ZFC is consistent, and let $T$ be the theory of ZFC together with all instances of this maximality principle. I claim that this theory $T$ is consistent. To see this, consider any finitely many assertions from $T$. This subtheory mentions only finitely many assertions $\sigma_0,\ldots\sigma_n$. Start from any model $M\models\text{ZFC}$, and then form extensions $M[G_0]\cdots[G_n]$, where at stage $k$ we force $\square\sigma_k$ over $M[G_0]\cdots[G_{k-1}]$, if this is possible to do while preserving $\omega_1$ and $\omega_2$, and otherwise use trivial forcing. The final model $M^+=M[G_0]\cdots[G_n]$ will satisfy all the instances in the subtheory, since if $\square\sigma_k$ is forceable over $M^+$ while preserving $\omega_1$ and $\omega_2$, then it would have been forced already at stage $k$ and hence would be true in $M[G_0]\cdots[G_k]$ and hence also in $M^+$. So every finite subtheory of $T$ is consistent, and so the theory is consistent, as desired. QED
Your theory follows from $\text{MP}(\omega_1,\omega_2\text{-preserving})$, since every forceable $\Sigma_1$ assertion is forceably necessary, because once you force it, it remains true in all further extensions. The parameters $\omega_1$ and $\omega_2$ are definable, by definitions that are absolute under $(\omega_1,\omega_2)$-preserving forcing, and so one doesn't need to have them as parameters, once we loosen the $\Sigma_1$ restriction to the forceably necessary statements.
This kind of argument is used extensively in my paper A simple maximality principle.
(Click on the edit history to see my original answer, which had some remarks on the Levy absoluteness theorem, which implies that $\Sigma_1$ sentences cannot be affected by forcing.)
