And this also happens when you don't have a RAM pack connected?
Sounds like a thermal-related problem. The best way to hunt for this would be using freezer spray - let the unit run until the faults appear and then spray each major component - regulator, ULA, Z80, memory IC, one at a time to see if that makes the fault disappear. Any electroyltic capacitors in the unit are also a possibility (spray those too).
Freezer spray made for the purpose of diagnosing thermal electronic faults is usually quite expensive but if you take a look at this thread, someone has just discovered more or less the same product being sold for an entirely different purpose, and much more cheaply.
https://vintage-radio.net/forum/showthread.php?t=164937
If you can bring yourself to buy some of that, that should work quite well for the purpose.