So this time, I decided to fix it myself. No experience in dryer fixing non-withstanding. That's what YouTube is for, right?
The first instruction on each and every page was unplug the dryer. Which I did, and ended up shocking myself so badly I was curled up in a ball behind the dryer for 5 minutes shaking. Now the dryer was still partially plugged in, and I was too afraid to unplug it. This was not a good start.
I did manage to get the dryer unplugged and shoved back into place. Two helpful little kids ferrying me screwdrivers, and twenty screws later; the top, and front were off, plus the drum. Aha, that's what a heating coil looked like, and they probably weren't supposed to be charred and in pieces. I believe I had found the problem.
One trip to Home Depot later I found out that you can't just go and buy these parts, they had to be ordered. I wonder how many women in their gnarliest clothes tow two kids into a hardware store asking for heating coils? I've gotta give people something to talk about. So for now the dryer will stay in pieces until Tuesday, and people can simply stop wearing clothes until then, or the clothes line will get lots of use.