Galaxy s III charging issue

I bought a GS3 during thanksgiving and it was working great. Couple of weeks ago, the charge would not stay. From 100% it would drop to 50% in new minutes. I had to switch off and restart the device to stop the drain. Also, the phone wont charge some times. I have to nudge the charge pin to start the charge process.

Couple of days later, I had the phone in my pocket and it started heating up. I could feel the heat on my thighs. I had to remove the battery for few minutes and put it back. Then I visited an AT&T store and the guy suggested that there might be a problem with the wall charger. I got a new charger and surprisingly it didn’t work at all. 

I called the device support and I was told the battery is the problem. They sent me a new one. The overheating stopped, but it’s still hard to charge the phone.

I am using  my original wall charger  (with difficulty in setting it up) and charging the phone. The same wall charger works for  a different HTC phone without issues (USB).

My car charger always works without any nudging. yesterday, I called the device support again, and I was told to factory reset my phone. I did that, but the wall charger takes very long than usual. 

Looking at all this, it appears there is a problem in the charge pin on the phone. I take good care of my phone.

I was told that I could get a replacement device and it’s only a reconditioned one. Not new. Has anyone seen this before? Any suggestions, recommendations?