Hi mc858! I had the same problem almost 2 years ago. I bought a new battery from
https://www.ipc-computer.de/asus/notebook/rog-serie/g7xx/g751x/rog-g751jy-t7060h/akku-10820707, but they don't have any anymore. This product is the original battery, but any aftermarket battery will be fine.
Just running it from AC is fine, I was doing it for weeks maybe and no damage was done. My only problem was as you also might have experienced, the power cable can slip out quite easily, but other than that, don't worry about it.
Also I was changing the battery for myself. It's quite a disassembling process, since you basically has to remove everything, even the hinge to get access to the internal battery... but can be done and then put it together in an hour or so, don't rush it. You can find instructions on how to do it on YouTube.