Have you checked the pins of the "unlit" LED's with a logic probe - there should be no signal when unlit (of course :-)). Then work back from there to see where there is a signal. Not as straightforward as it sounds because the displays are multiplexed, but you should get an idea of where the problem lies.
I'll assume the PIA is working as other functions are ok, so check IC6, IC9 and IC10, plus transistors T1-T7, but from what you describe I'm going to guess at a problem with the output of IC10 reaching the LED display.
On mine, all IC's are socketed so hopefully yours are too ?