Just a few thoughts.
1. 96% is not uncommon. When I have my iPad Pro 12.9” (1st gen bought when it came out) on power for days on end, it will charge to 96% and stay there. If it’s been out for the day and it is down to 20% it will charge to 100% again.
2. The power adapter might get hot. Really hot when it is charging at full power. I have had it many times, it still worries me, but I haven’t damaged one yet with it. They seem to be ok with it.
2.b. If you want faster charging, buy a 29W USB-c and a USB-c—>lighting cable. Bloody expensive, Bloody fast charging. If you use a 12W charger it might discharge faster under heavy load then the 12W can charge. I certainly thin that on the 12.9 the 29W charger should have been standard. Both the 10.5 and the 12.9 now can fast-charge with the 29W. I’ve bought 3 of them (two at home for charging both the 12.9 and 10.5 and one for quickly topping up the road).
3. It really looks like something on your device got stuck. It’s almost always something software related. It’s almost always some app that does this.
4. With your new iPad, restore it, use it like you did before.
5. If you get the same behaviour, completely erase the iPad, restore to factory settings and don’t start from a back-up, just download any app again from the App Store, just like you’re a first time Apple user. Usually iCloud data doesn’t effect it, but all the settings, from mail, apps, reminders, localisations services etc. Etc. Should be entered again. It is a pain in the proverbial place, but might be the only way to get rid of it.
My experience with Apple has been great so far, but I think your phone support experience has been substandard. Part of the problem is they keep extensive records of your device/case and when the first decided it was a software problem, you’re basically stuck with this.
I seriously hope this new device will solve your problems. the 10.5 is a wonderful device and IMHO the best tablet there is!
Full sized keyboard according to my fingers, super nice screen, ProMotion, huge disk size (512Gb) 4Gb ram, really bright, good battery life (when you don’t put it on 100% brightness

) blistering fast and big enough to write well on. Just like the paper notepads I used back in the days.