After running either just the central HVAC fan, or the central HVAC AC for just a few minutes, the Bryant thermostat (Model TSTATBBPB101, C Wire attached) screen momentarily shows "PF", and then goes blank (it normally shows the current and target temperatures).
1 minute later (if running just the fan), or 5 minutes later (if running the AC), the thermostat's screen comes back on, and everything looks good.
The fan (or AC) will then turn on, and work fine, until the above cycle repeats after just a few minutes.
I then installed a new LUX TX9600TSa thermostat, and the problem persists. Without batteries, but with C wire connected, the thermostat will show "Low/Battery Low" after running the AC for 5 minutes, and then go blank. With batteries installed, and the C wire still connected, the fan (or AC, if it is selected) will simply stop after a few minutes, even though the fan is still turned on (or in the case of the AC, the target temperature has not been reached).
1 minute later (if just running the fan), or 5 minutes later (if running the AC), everything will reset and look fine, but the problem persists: I can no longer get the HVAC fan or AC to run for more than a few minutes at a time.
Just to ensure it's not a clogged filter, I put a brand new MERV 16 filter into the HVAC system, but the results did not change.
What is likely the problem, and is this something I can fix myself?