"I also have a Carrier furnace model 58GS050-3 but no central air, motor will not shut off. If I set the thermostat to 'system off' the fan keeps going. Everything works fine including the burner (flame) that goes 'off and on' when elected at the thermostat. I have to remove the lower door to activate the safety switch or unplug the furnace to turn the blower off."

This can be several things. The most common reason is the circuit board. That's funny, I just left a home with that exact problem and it was the circuit board. Make sure that the fan switch is set to auto. If the problem is still there then disconnect the thermostat wires at the furnace.When you do that note where each of the wires go so that you can rewire it later. Plug the furnace back in and check to see if the fan is still on. If its not then you have a short in the thermostat wire. You will need to replace the thermostat wire or repair the damaged part of it. If its still on then more than likely your circuit board is gone bad.
Hope this helps!

I was going to figure this out through you guys' help and replace any bad part buy purchaisng it at the internet but I don't know how to locate which part is bad even with your suggestions, it sounds too complicated for me. So, I have an expert come over and he found the circuit board to be bad, you are right. The cost to replace the circuit board compared to replacing the whole unit (30 year old), I decided it is more feasible to install a new furnace. Thank you for your time, it is very much appreciated.