How to find the Heisenberg picture from the master equation without Lindblad structure?

My question is closely related to this post and this one too. I understand that with a Lindbladian type master equation, it is possible to find the differential equation for an observable. However, my master equation looks like this:
$$frac{d}{dt}rho(t) = -frac{i}{hbar} [H,rho(t)]-f(t)[q,[q,rho(t)]]+g(t)[q,[p,rho(t)]].$$
Firstly, does the argument from the answer in the first linked post hold here? I’m not too sure if the caveats raised in the comments apply here.

Secondly, if we describe the above as $frac{d}{dt}rho(t) = mathcal{L}[rho(t)]$, then how do I find $mathcal{L}^{dagger}$, so as to find the Heisenberg picture? Of course I’d know how to find $(mathcal{L}[rho(t)])^{dagger}$, but I’m not clear on how to get the adjoint of the superoperator itself.