DIY's frequently assume it is always the thermostat that does the dastardly deed; because it is the most visible representative of the heating/cooling machinery. It is like a politician, it starts things; but hasn't a clue what happens after it starts things; but it can stop things by cutting off supplies (of electricity.)
The most likely cause of your problem is a bad fan switch that has had its range of operation stray below room temperature. Adjustment or replacement might be needed. A thermostat will not start furnaces doing things out of the ordinary unless they have been newly changed, but a fan switch will.