I have heard of a recent common issue with the battery going bad on these tablets, so it is very possible that you simply have to pop off the back, which is held on with very strong clips, lift a few flex cables and swap batteries, but since I am just guessing what your exact issue is, you may also have a damaged charge port. If it feels loose when you plug it in, you may need to replace that instead, which if it is attached to the motherboard, you are going to need soldering work, possible, just higher risk than a battery swap.
When the digitizer and LCD actually separate from the plastic frame, the best thing to do is remove them from the device, and with the strongest and thinnest adhesive you have, reapply the screen to the rest of the frame, give it time to set before reassembling the phone as well, these repairs are simple, but if you're not sure what you're doing, a lot can easily go wrong.
Though the LCD isn't broken on your device, it looks like it is defective, if you have experience taking the S3 apart, and access to an additional S3, you could simply switch motherboards to test it. Honestly, it would be best for you if it is the screen, because the only other possibility is that your main logic board is having issues, which isn't good at all, in that case you would have to get a new phone all together.
It sounds like you may have a bad battery on your hands, it isn't terribly difficult to replace, though if you don't have much experience opening them up, I would advise bringing it to a reputable repair shop and having experienced technicians look at it for you to be sure.