Let $A$ be a commutative ring (not necessarily Noetherian), $I=(f_1, f_2, \dots, f_n) \subseteq A$ a finitely generated ideal that is weakly proregular, or better yet, generated by a regular sequence. Let $M$ be a derived $I$-complete $A$-module (which means $\mathrm{Ext}_A^{i}(A_{f_j}, M)=0$ for all $i\geq 0$), not necessarily finitely generated. Assume further that $A$ is (derived or classically) $I$-adically complete, and suppose further that $M$ is $I$-completely flat (sometimes also called $I$-adically flat), meaning that $\mathrm{Tor}_{>0}^{A}(A/I, M)=0$ and $M/IM$ is a flat $A/I$-module.

My question is

If $x \in A$ is a non-zero divisor of $A$, is $x$ necessarily a non-zero divisor on $M$?

This seems true at least under some further assumptions on $x$, such as $xA \cap I^k=xI^k$ for all $k$, but I am actually interested in the somewhat perpendicular case when $x \in I$, or better yet, the general question whether such $I$-completely flat modules are torsion-free.