Why is my thermostat reading falsely high temperatures?

Yes thermostats do go bad. The old double mercury thermostats are great for simplicity and no batteries needed.

Yes digital thermostats do fail. Just like all electronics they have a life and if in an electrical noisy area the life will be shortened. I have only seen a few thermostats that were actually integrated as part of the heating / cooling system.

Depending on the features your system has you will need a common wire a call for heat and a call for cooling, fan only, different fan speeds / heating levels . You can install the old thermostat and although it won’t cost anything today loosing some of the features of a modern digital may be worth the expense. Most only have cooling/ heat a fan only setting and off.