Why is my phone rebooting and not charging when plugged in?

[AC Question]

I have a Motorola XT1095, purchased used about 2 weeks ago. Last night, I plugged it into the car charger when it was at about 10% battery and it instantly powered off as though out of batter entirely. Since then, when connected to any charger (including the factory Moto charger and a Samsung charger I've been using for years without incident), the phone screen lights up and displays the battery symbol as per normal charging, then the phone vibrates and between 3 and 10 seconds later, the screen goes black and the cycle repeats. I have had the phone plugged in for up to 30 minutes at a time with the phone power cycling continuously.
Phone worked flawlessly until this incident. I'm assuming that some wiring in my car's charge port is messed up and somehow caused a power surge in the phone, or else drained the battery beyond its safe limit.
Trying to figure out if changing the out the charge port is a reasonable response. Phone's warranty is expired. I'm fairly handy electronically and I really cannot afford a new phone, so any insight into potential causes of this problem/potential fixes would be welcome. Thanks.

 

[Rukbat]

1. The phone could have gotten wet inside (condensation counts). The charge-buzz-restart cycle is typical of a wet phone.

2. Bringing the battery down to 10% regularly may have killed it. You should try to never bring a lithium battery down to less than 40% (50% is better). At 10% you get only a few hundred recharges before the battery dies.

3. Changing the port is usually easy - getting to it to change it may move something that stops the phone from working. "Handy electronically" and "familiar with fixing cellphones" are entirely different things.