After chasing down references relating to the paper of Shelah mentioned by Will Brian, I now have a satisfactory answer to the question.  It all hinges on whether there is a splitting of the quotient algebra ${\mathcal F}/{\mathcal N}$ of the $\sigma$-algebra ${\mathcal F}$ by the null ideal ${\mathcal N}$, that is to say a Boolean algebra homomorphism $\Phi: {\mathcal F}/{\mathcal N} \to {\mathcal F}$ that is a left inverse for the quotient map $\pi: {\mathcal F} \to {\mathcal F}/{\mathcal N}$.

First suppose that such a map exists.  Then for each $\alpha \in A$ and $\omega \in \Omega$ there is a unique element $\tilde X_\alpha(\omega)$ of $\{0,1\}$ with the property that
$$ \omega \in \Phi( \pi( X_\alpha^{-1}( \{\tilde X_\alpha(\omega)\} ) ).$$
It is a tedious but routine matter to check that $\tilde X_\alpha: \Omega \to \{0,1\}$ is a modification of $X_\alpha$ (a random variable that agrees almost surely with $X_\alpha$), and that the $\tilde X_\alpha$ satisfy every sentence $S \in {\mathcal S}$ surely (rather than just almost surely).

Conversely, suppose that every family of random variables $X_\alpha$ that almost surely obeys each sentence $S$ in a family ${\mathcal S}$ can be modified to surely obey such a sentence.  We consider the family $(X_\alpha)_{\alpha \in {\mathcal N}}$ defined by
$$ X_\alpha(\omega) = 1_{\omega \in \alpha}$$
and consider the Boolean algebra homomorphism sentences
$$ X_{\alpha \cup \beta} = \max( X_\alpha, X_\beta ); \quad X_{\alpha \cap \beta} = \min(X_\alpha, X_\beta )$$
$$ X_0 = 0; X_1 = 1 $$
$$ X_{\alpha^c} = 1 - X_\alpha$$
for $\alpha, \beta \in {\mathcal F}$, together with the sentences
$$ X_\alpha = X_\beta$$
whenever $\alpha,\beta$ differ by a null element in ${\mathcal N}$.  Then the indicated random variables $X_\alpha$ obey each these sentences almost surely. By hypothesis, there exists a modification $\tilde X_\alpha$ of each $X_\alpha$ that obey these sentences surely.  If we then define $\tilde \Phi: {\mathcal F} \to {\mathcal F}$ by the formula
$$ \tilde \Phi(\alpha) := \{ \omega \in \Omega: \tilde X_\alpha(\omega) = 1 \}$$
then one can verify that $\tilde \Phi$ is a Boolean algebra homomorphism such that $\tilde \Phi(\alpha)=\tilde \Phi(\beta)$ whenever $\alpha,\beta$ differ by a null element, and such that $\tilde \Phi(\alpha)$ differs from $\alpha$ by a null element.  Thus $\tilde \Phi$ descends to a splitting of ${\mathcal F}/{\mathcal N}$.

As mentioned by Will Brian, the main result of 

<cite authors="Shelah, Saharon">_Shelah, Saharon_, [**Lifting problem of the measure algebra**](http://dx.doi.org/10.1007/BF02760673), Isr. J. Math. 45, 90-96 (1983). [ZBL0549.03041](https://zbmath.org/?q=an:0549.03041).</cite>

is that it is consistent with ZFC that $[0,1]$ with the Borel sigma-algebra has no splitting; on the other hand it is a classical result of von Neumann and Stone that assuming CH, this measurable space has a splitting.  So for this space at least the problem I asked is undecidable in ZFC!  On the other hand, the main result in

<cite authors="Maharam, Dorothy">_Maharam, Dorothy_, [**On a theorem of von Neumann**](http://dx.doi.org/10.2307/2033342), Proc. Am. Math. Soc. 9, 987-994 (1959). [ZBL0102.04103](https://zbmath.org/?q=an:0102.04103).</cite>

shows that a splitting always exists for complete probability spaces.