Unexpected ShutDown

i have an old MacBook Pro early 2011, I love this machine and Even though I have newer models I can’t let any of my old Apple products go. I have recently had a problem with it where it would unexpectedly shut down. At first I was positive that it was a battery temp sensor that was causing it, I ran diagnostics and it popped up with a temp sensor error but then disappeared and hasn’t returned. The issue is still occurring though Even though the MacBook is saying there is no issues with the sensors, up on closer inspection when the MacBook is running of battery power the battery connector end that connects the battery to the logic board gets extremely hot but when connected to AC power adapter it doesn’t. Also bearing in mind that when the MacBook is connected to an ac power adapter the random shutdown don’t occur. Could this not be a faulty temp sensor but the battery? Could it be putting out to much power or overheating and the temp sensor kicks in because it’s to hot and shuts itself down to protect itself? This would explain why it doesn’t happen when it’s connected to ac power! Please help with all suggestions! Thank you.