Sounds like you do have two different issues here.
The power adapter and its cord, or the connector on your system is not making good contact. Clean the contacts and make sure the power adapter contacts & cord are in good shape. If not you'll need a new unit.
The battery is showing it's age here. Once you get the AC power side of things stabilized. You could try doing a deep discharge of your battery and reset the SMC. I must warn you here sometimes when you do this the battery may not recover. You'll then need a new battery.