Does a finite ring's additive structure and the structure of its group of units determine its ring structure?

Let $A$ and $B$ be finite commutative rings with unity. Denote the additive group structure of each to be $A^{(+)}$ and $B^{(+)}$, and the multiplicative group of units of each to be $A^{(\times)}$ and $B^{(\times)}$ respectively. Supposing that $A^{(+)}\cong B^{(+)}$ and $A^{(\times)}\cong B^{(\times)}$, does it follow then that $A\cong B$?

No. Suppose that $p$ is an odd prime, and consider $A = \mathbb F_p[x,y]/(x^2,xy,y^2)$ and $B = \mathbb F_p[z]/(z^3)$.

In each case the addive group is isomorphic to a direct sum of 3 copies of a cyclic group of order $p$, while the multiplicative group is isomorphic to a direct product of a cyclic group of order $(p-1)$ and two cyclic groups of order $p$.