Can the embedding theorem be improved to preserve injective objects or even better be part of an adjoint pair? Or if that's not possible are there stronger conditions (AB5, enough injectives, etc) on the abelian category in question that would suffice?
Being explicit, if $\cal{A}$ is a small abelian category is there an exact embedding $F:\cal{A} \rightarrow \mathbf{MOD}_R$ that preserves injective objects?