Have you tried other lighting cables first, just to check if it’s getting a charge at all? The battery will naturally drain, albeit very slowly.
Regarding changing the battery. Yes, not having a full charge is best, but you don’t have to have it fully drained when replacing it. Follow the steps and take your time. As long as you SLOWLY work the adhesive from under the battery, the battery comes right out with absolutely no force.
What I used to do is get a pair of needle nose tweezers and pinch the black tab at the end of the adhesive. Then pull a little of it from under the batter and twirl the tweezers. I’ll grip on a little better and give you more of a handle to work with.