Old capacitor leakage either in via holes causing shorts or old capacitor leakage got underneathe various components causing shorts.
For via holes, the solution is to do what you did to fix your pulled up pad. Run a solder lead through said hole and solder to both sides.
For leakage under components causing shorts, you have to de-solder said part from the board, clean the board and part, then re-solder back to the board again.
There is the minimal chance that it's actually the screen at fault, but that's extremely unlikely.