My dryer was doing the same thing, stopping randomly, starting again, pressing start button and display would show "drying" but the dryer never started. As luck would have it my wife's solution was to give the door a "slam" OK, a few slams and the dryer would start. This got old really fast, and her method of starting the stubborn dryer got less and less effective.
Needless to say the problem was at best intermittent. I thought it was the door switch, but proved it was fine by holding the switch "in" while I started the dryer, both when it worked and when it only said it was "drying"..well, that would have been too easy anyway...
So off comes the cover, power board & control panel in view, both suspect at this point, find the yellow paper, check the wiring schematic, kinda Greek and all.
So in my poking around in there I bump the wires that go into the 4-pin Molex connector (mid power board, all the way to the right) and lo and behold the dryer starts! Well look what we've found, a loose wire connection to the board! Kinda explains the randomness of the problem and how the vibration from my wife's solution worked, wiggle the wire(s) and it would "start", also wiggle the wire(s) and it would stop.
What to do? GE doesn't sell wiring parts or a harness, plus I need the special tool to remove and re-crimp new female pins from inside the molded plastic "plug". I tried splaying the ends of the male pins soldered to the board with an o-ring pick, seemed to work, closed the dryer up ran great for a while, then more of the same, wash-rinse repeat...
I checked the board-mounted pins for cracked solder, ohmed them out, all was fine, simply some worn out female connectors in the Molex plug. Surely a low $ item if you can get them..
For the minute, I've left the cover off and will jostle the wires to get her going...Just an observation, but the plugs are considered "vertical" plugs with the wire leaving at a 90 degree angle, really could have used an inch or two more wire so as not to make the bend so sharp thus stressing the pins less.
At this point a replacement harness (used or new) would be ideal, but I guess I'll have to bite the bullet and get the right tools for the job. Any suggestions would be greatly appreciated...sorry this was so long, but I think it will help someone. :)