I haven't had any electrical problems with my 06GT yet (had to take it in for a rear brake shake thing), BUT... my Jeep Liberty (ok, wife's) started acting weird. She would get the low gas warning and it would display like 3 or 4 gallons with a quarter of a tank. The alarm would go off when she set it with the key fob. It would stumble on startup... etc.
Well, there was a LOT of corrosion buildup on the negative battery terminal. Took the leads off to clean them up and found a crack near the negative post on the battery. Replaced the battery, and no more problems.
Hope that gives you something to check on... Here's some websites to check out that help me...
http://www.trustmymechanic.com
http://www.faqfarm.com/Q/FAQ/470