That battery is toast, you will have to replace it.
It is fine to run the machine on ac often. It is advised to run down a laptop battery completely every once in a while though. I try to run down my thinkpad at once or twice a month. I don't think I would run down the battery constantly. Both running on ac all the time and draining the battery all the time will wear it out faster. Just run it down every so often and other than that use it how ever you want.
The battery should be very easy to replace. There will be some sort of realease switch and it should slide right out. Then just slide the new one in. The switch will probably have a symbol of a padlock both locked and unlocked.
I'd check for a battery on ebay, it will be A LOT cheaper.