Hi and welcome! Yes, your heater fan should not run constantly. Mine comes on a bit before the heater starts producing heat and then runs for a bit (to extract all the remaining heat) after the furnace fire goes out.
It's possible there's a fault in your thermostat. Several members have replaced the cheap mechanical OEM thermostat with an inexpensive, battery powered (digital?) one from Wall Mart. The replacement does a better job of holding the desired temperature too, which was their reason for the switch. You might try that as a possible solution. I think the brand name of the new thermostat is Honeywell. If you do a search in the forum, you should be able to find a few threads on this conversion.
- Jack