In the last week, the draft inducer motor in my Rheem furnace has begun to periodically kick on and off for a few seconds at a time. This is occurring while the thermostat is in cooling mode. The air conditioner is working fine. The furnace is nearly 20 years old. The draft inducer fan/motor was replaced about 10-12 years ago after the bearings went out.

Sounds like the control board in the furnace is going bad. Check all the connections on the board, make sure wires are not too close to each other.