I have a similar issue to this. I was able to set jumper in the furnace
to cycle the air longer after a heating stage, but it doesn't run it
long enough.In your case and mine I would like the ability to sync two
Nest thermostats. In your scenario so ...
Most likely a sensor in the base of the thermostat has failed. Yes, it
surprised me too, there is circuitry in the base that has sensors in it,
I found a refurbished base on eBay and now I am not having this issue
anymore.