Whatever you decide to do, do NOT fully discharge the battery like a NiMH one, hoping to erase the "memory". Li-ion batteries do not have "memory" and every time they're fully discharged they lose some cells, so they will have less capacity. Eventually, the capacity will decrease to the point where they won't be able to keep a full charge, like it's already happening. my advice, use it plugged in and if you need it to work reliably unplugged just get a new battery from a reliable source (not some shoddy seller that may give you a used/shot battery).
Just my 2 cents...