I have a Kenmore Elite dryer, 6 years old. The dryer will only turn on in the auto-moisture setting and then quickly speeds through and shuts off. Won't operate on the normal timer setting

Sounds like your timer is messed up. If that's all its doing, Id try that first. After all, the fact that it will run in the auto moisture setting, but only speeds through it to quickly suggest to me that everything is working.