My first thought was the sail switch is sticky. But now that it's running on the 15A fuse I would guess the grease on the furnace motor is dried up. You'll probably have to replace the motor in the end to get it working.
Is there a name and model number on the thermostat? Does the thermostat control both the heat and AC, or just the heat? You should be able to pull the cover off and see if there is any obvious adjustment. But it sounds like the thermostat is acting up and I would just replace it. You can buy an inexpensive digital thermostat to replace the original. I bought mine for $25 from Walmart. If you go this route just make sure the thermostat can run off batteries and supports a 24v system. The swap for me was easy. It took more time drilling two holes in the wall than it took to wire it up.