I think not, my argument is slightly messy but it works I guess.

Take the Hirzebruch surface $S=\mathbb{F}_2$ with its standard toric action, it is diffeomorphic to $S'=\mathbb{CP}^1 \times \mathbb{CP}^1$, but I claim they are not equivariantly diffeomorphic (with respect to the standard toric action on $\mathbb{CP}^1 \times \mathbb{CP}^1$).


Suppose that they are, so $F: S \rightarrow S'$ is an equivariant diffeomorphism.

Then, pick a subtorus $S^1 \subset T^2 $ such that the corresponding Hamiltonian $S^1$-action on $S'$ has an isotropy sphere $C'$ with $C' \cdot C' = 2$ (it will be one of the sections of the $\mathbb{CP}^1$-bundle). Then, there will be a corresponding subtorus action on $S$, also having an isotropy curve $C$ with $C \cdot C = 2$. But the isotropy submanifolds for subtorus actions on toric surfaces are always the boundary divisors so $C \cdot C = 0$, a contradiction.