Furnace heats and runs for an appropriate amount of time, say 5-10 minutes depending on weather. After this running phase, approximately 2 minutes later, the furnace blower (no gas ignition) turns on and runs for 30-45 seconds. It does this for every cycle.

I had a service call and they checked the furnace but found no problem. They did manipulate the furnace to run and when they stopped it, it did not over-run.

The furnace is 18 years old and this started last year. The thermostat has been replaced to no avail.

Sounds like your fan ON temperature is set too low or the fan OFF temperature is set too high. One way or another, residual heat from the exchanger is making the fan turn on. I'd set the fan ON temperature a bit higher and the fan OFF a bit lower. The fan switch is located inside the heater on the exchanger casing.