Definitely get the battery checked out. Like previously mentioned AutoZone will will do it for free. The new handheld AutoMeter battery testers many Autozone stores are now using are quick and accurate. Even though your battery is a "12 volt" battery, 12.4 volts is roughly a 60% charge. 12.7 volts + or - .1V is a full charge. These modern day automotive electrical systems are notorious for doing funny things when the battery drops below 12.5 volts. Domestic Chrysler vehicles and European cars are well known for it, and ours kinda being both of the two doesn't help the matter much!

Remove the battery and have it charged and load tested. If it tests out good clean any corrosion from the battery cables and apply di-electric grease to the terminals and cable ends before reinstalling. Make sure the terminals nice and tight. I believe our OEM battery group size is a 94 series. Correct me someone if I am wrong.

If you are still having trouble take it to Chrysler!