after looking a little deeper into the schematic and the Pin definition of the Heltec Module it looks like the Touch functionality is defined inside the µC.
So not only a short can disturb the function but also a leakage current on the pins 2 and 12. After checking for shorts (as you did), I would check for the resistance between the pins and GND ( board not powered). Having a 2nd set, you can also check if the change of the Heltec module will cancel the problem or it stays.
If nothing is wrong with the PCB and the change of the Heltec module cancels the problem, the only try is to clean the module (way of pins to the µC) with PCB cleaner (mostly alcohol).
I do not believe, that the power supply can disturb the function: I was running the module without battery from the USB port of my desktop without issues.
vy 73 de Karsten, DD1KT