I would consider replacing the battery.
I had similar battery issues, my iPhone 6 would shut down randomly and the battery would drain quickly after a full charge. Had to keep the phone on the charger constantly!
Ordered the iFixit battery replacement kit, and replaced my battery today today. Phone is working great so far.
My suggestion is to watch the battery replacement video and follow the step by step instructions to the T, if you are planning to do it yourself.