My son had the same problem. Battery started to fully charge to only 80%, then the phone began randomly shutting down. He asked me for advice.
I decided to take the battery out.......... which proved difficult. the battery had swollen in the casing. Eventually got it out with some tweezers and the battery had swollen a fair bit.
Got a replacement battery and its working a treat. So much so he has now racked up a £107 bill.
Your damned if you do, you're damned if you don't!!