The counit map is cocontinuous in M, so using the fact
that any cofibrant object is a retract of a transfinite composition
of cobase changes of generating cofibrations of A-modules,
combined with the left properness of the model category of A-modules
and the fact that both adjoints preserve monomorphisms,
the problem boils down to showing the claim
for the case when M is a domain or codomain of a generating cofibration.
In this case, this amounts to M=A[n] or M=(A[n−1]←A[n]),
and in both cases the claim is trivial.