** If indeed I do need a new motor, are there any other parts that I should order that go with it? I'm going to order a new belt also, so I won't have to take the dryer apart again in a few years when that wears out. **
It depends on the design of whatever model you have. Parts that often eventually need replacing are the idler (belt tension) pulley, drum rollers/slides and possibly the blower fan itself (it can get loose on the motor shaft). I don't know if I'd replace any unless there was some sign they were defective or were worn a little. The belt is a good bet though regardless.