Here are your parts
Parts for Frigidaire FSE747GES1 Dryer - AppliancePartsPros.comCould be that the Start Switch contacts are welded together.
Unplug the unit and both wires at the start switch.
Then measure across the start switch with a meter.
Should be infinite ohms.
Should be o 0 ohms when activated (pushed or turned).
If the switch is OK then odds are that it is the electronic control board.
I am doing a bit of guessing here as I could not find a decent copy of the wiring diagram.
I doubt it is related to your heating element as the element circuit is separate from the motor circuit.
As an aside be sure to lubricate the rrear bearing with high temperature greas when reinstalling it.
It should be safe to run this way.
But as usual there is a but and in this case the but is that it will probably blow the door switch in time.
If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.
4. When you start always short the meter leads together. This will tell you that the meter is working and if there is any 0 offset.
There is a good STICKY at the start of this forum about it's use.
I am off to the lake later today so will not be available till next week.