One of my post war 38/44 HD's has the problem that when the gun heats up from firing the trigger does not reset unless you flick it out with your finger. If you loosen up the screws it works fine though.

I took it apart and found that the interior is a bit gunky from long term rust. I cleaned the bulk of that out but it is still a bit rough. This is an old cop gun, so I bet that somewhere along the way it go water inside.

Any suggestions or thoughts?

Take it apart and polish both the rebound slide, and the area the rebound slide rides on in the frame. Since you'll have those parts out anyway, replace the rebound spring with a 13 to 15 pound version from Wolff.

The trigger rebound spring is too weak. Factory springs are rated for 18LBS. Perhaps someone cut or replaced the one in there now.
Try CLP and a tooth brush to help kill the rust. Good luck