Let $M$ be an $A$-module. Is its injective hull affected by whether I regard $M$ as an $A$-module or $A/\mbox{Ann}(M)$-module ?

-
Yes, take $A = k[[x]]$ and $M = A/(x)$. Then as a $k = A/(x) = A/\text{Ann}(M)$-module, the injective hull of $k$ is $k$. As an $A$-module, the injective hull is much much bigger.