I’m sorry, I misunderstood that. If your vibration works otherwise, then the module should be OK.
Which battery guide did you use? This one does not mention vibration. There might be other suggestions in that guide which are worth a try.
The rest of the story sounds like a broken battery. I’ve seen that old batteries can suddenly start to discharge quickly when a power-hungry app is opened - probably caused by a sudden drop in voltage. Your problem is similar but more extreme.