The blower on my furnace doesn't stop running.

call a furnace repair service locally

Make sure your air filter is clean. Is the thermostat satisfied? or is the room temp. still below the thermostat setting? When ever you have to turn the power off and then back on to a furnace that usually indicates that the furnaces ignition control locked out. The first thing that I would try is cleaning the flame rod.

The reset button near the ignitor is a high temperature limit safety switch. When the furnace gets too hot and the manual reset limit switch is tripped, it most likely goes into a 'safe' mode where the gas control is locked out electronically and the blower is made to run continuously to rid the heat exchanger of the excessive heat. This will continue until the reset is pushed and the power is cycled off. It will remain an endless cycle unless the reason for the overtemperature condition is not corrected.

Your furnace 'temperature rise' is getting too high due to a lack of air across the heat exchanger for whatever reason. Some of those reasons could be a restricted air filter or a blower motor that's cycling improperly on its internal winding thermostat. You could also have a dirty air conditioner 'A' coil (evaporator) on top of the furnace restricting the air flow. Nevertheless, you are going to have to have it serviced by a reputable service company's technician to correct what's wrong.

Sounds like the fan limit was not replaced correctly. You can try switching the blower to the On position (instead of Auto), if the furnace runs properly with the blower on that would be the most likely culprit.

Flame sensor needs to be cleaned. turn off the furnace power switch or breaker. pull the flame sensor located in the burner runner section of the furnace. It will be easy to locate because it has 1 wire running to it. pull the screw being very careful not to harm anything else. get the sensor out. inspect it for cleanliness. clean the sensor with light grade sand paper gently. re-install the sensor plug the wire back on. restart your furnace and this should fix the problem.
CookHVACR

Sounds like fan limit switch/fan timer

This will cause the main burner to run untill the over tempreture switch is tripped therefore shuting gas supply and running fan on high till over heat condition is resolved then it will restart burner and repeat
continuing to run heater like this will cause heat exchanger split.

It is possible that you did not set the new blower on the proper speed. Black wire is high. Blue is medium. Red is low. If the fan is operating to slow it can run to hot and trip the high limit control. Make sure the filters are clean. When the old motor went bad, the problem might not have been discovered rite away your high limit control might need to be changed.
Also anytime you work on a furnace you can kick up some dust. Always clean the flame sensor.
Good Luck
HVAC Teacher

replace the circiut board or the blower relay. It will probably a board.

YOU COULD EITHER HAVE A THERMOSTAT PROBLEM OR FAN CONTROL PROBLEM.TURN THE POWER OFF AT THE FURNACE AND TAKE THE THERMOSTAT OFF THE WALL.MAKE SURE YOU MARK THE WIRES SO YOU KNOW HOW TO HOOK IT BACK UP AGAIN.RESTORE POWER TO THE FURNACE.TOUCH THE WIRES THAT WERE HOOKED TO THE R & G TERMINALS ON THE T-STAT TOGETHER AND HOLD THEM THERE.OBSERVE THE OPERATION. WHEN YOU TAKE THEM APART DOES THE BLOWER STOP?TRY IT A FEW TIMES. IF IT CONTINUES TO START AND STOP YOU GOT A BAD T-STAT.JEFF