Hi, the Galaxy Note 2 charge port is fairly simple to replace, being a simple flex cable, and will probably solve the issues that you're having. A port can charge, or charge slow, or even appear to charge, and then not, when you have a damaged charge port, all kinds of bad things can happen. I highly recommend that you stop charging it, and replace that charge port assembly. If that doesn't work, then it is time to look at the power management IC, that's a little more of an in depth repair, but not impossible. Hopefully it's just the charge port!
I hope this helps you out.