When you cleaned your carb what all did you do? How far into it did you dig in? Most mechanical fuel pumps will put out around 7psi. This is even a hair high for most edelbrocks.
I recently swapped my old worn out 350 for a 400 that I built. The new fuel pump i installed was putting out about 7.5 psi. I went with a built Qjet from Quadrajet Power and after doing some research I found that Q-jets don't "like" more than about 4.5 psi.
The fact that you can turn the screws all the way in and stay running really bugs me. There is definitely something wrong in your carb. Either too much fuel pressure is overpowering the needle/seat and float, or you have a bad float, or just a bad needle and seat.
I am a long long long ways from a carb expert, I only got into the carb game a couple years ago. I'm sure there are others on here with some better advice, but if I were in your shoes, I'd get that carburetor straightened out before I pissed with anything else. You may in the end find that the carburetor was causing your issues all along.
Another thing you might do is verify your vacuum readings with another gauge if you haven't already done so... Wouldn't be the first time a gauge went south..