I'm a bit out of practice with these types of questions, and I've never really worked with $S$-modules, but I think the answer is yes. Here's why I think so. First, it's sufficient to check it for the case when A is a (co)domain of a generating cofibration in CAlg(R), by a standard cellular induction (for details, see Theorem A.2 in Hovey's paper on Smith ideals). Thus, we can assume A has the form $Sym(B) \wedge R$ where $B$ is a (co)domain of a generating cofibration of $S$-algebras. Thus, $A\wedge_R -$ is a weak equivalence of $R$-modules if and only if $Sym(B) \wedge -$ is a weak equivalence (in the underlying category of $S$-modules). Since (co)domains of the generating cofibrations of $S$-algebras are cofibrant (see MMSS), $B$ is cofibrant. Thus, $Sym(B)$ is cofibrant as a commutative $S$-algebra. Next, Lemma 3.7 of my paper on commutative monoids (accepted to JPAA), proves that $Sym(B)$ is cofibrant as an $S$-algebra (so the EKMM result you cite finishes the proof). I hasten to note that it's not recorded in print anywhere that the category of $S$-algebras satisfies the strong commutative monoid axiom, but for the crucial place in the proof of 3.7 where this is needed, you can use Proposition 4.2 of Shipley's A convenient model category for commutative ring spectra instead. Basically, the idea is that it's good enough to work with positive cofibrations to get the result you want. The last section of chapter VII of EKMM is also relevant, and could perhaps avoid the shift into positive cofibrations.