By interpolation $e^{tA}$ will be bounded on $L^p(X,\mu)$ for $2 \leq p \leq \infty$. Then the dual of $e^{tA}$ will be bounded on $L^p(X,\mu)$ for $1 < p \leq 2$; but since $A$ is self-adjoint this is just $e^{tA}$ itself. (I think there are no problems with this argument but I'm not 100% sure)

I'm not sure if you can say anything about boundedness on $L^1$. However, I haven't used the assumptions of finite measure or contractivity, so maybe these can be used.