The dryer terminal is something I see frequently and is usually nothing more than an improperly installed cord during installation. If you want to check the motor, there are 2 ways to do this:
1. Check the resistance of the motor at the run terminals, #5 (yel) and #4 (blue). There should be approx. 2.25 Ohms.
2. Or you can do an amperage check by applying a test cord to these terminals to look for about 5.3 amps while it is running. An easier way to check it would be to temporarily repair the black wire at the terminal block, then place your ampprobe on the black wire and start the dryer on an Air Dry cycle (no heat). This way you can read only the motor draw current.
The washer problem sounds more like a failed touch pad on the control panel. As a precaution you should also bid a new control board in case it was damaged by the shorted panel, however it may not be necessary to replace it once the control panel is replaced. Sometimes you get lucky that way. Knowing you can return a part that is not needed is good for you in this case. You'll find a thick tech sheet inside the control panel which is useful.