Even with AC adapter plugged in player seems to only run off the battery? Why does the battery need to be fully charged if I have the adapter plugged in?

Because the power supply, is just a "Trickle Charger" for the battery, and cannot, deliver enough current to run, the entire thing, alone, only from the wall unit. If however a higher current adapter could be used, no doubt you could use it just on that. You would need a switchmode type supply, something that has same plug, polarity, plug can be "Grafted" on to a new supply, this supply needs to provide the same Voltage as written on the PSU, but with a higher Current, Amps, draw rating, say 3 - 4 Amps. Then that would operate the unit alone. Without the battery. You are meant to use two batteries, one charge and one in use, and swap them out when needed to.