First take a deep breath.
So your axim wont charge unless you take the battery out?
And it wont sync? But it still works fine?
If so, it sounds like something may be wrong with your sync connector. Maybe it shorted out, who knows. But chances are that port it not functioning.
Second, the wireless button enables the last used wireless source, whether it be bt/wifi.
Try going to the wifi icon, it looks like cell phone signal bars, but its broken greyed out, except one bar is red.
Refer to the pic.
Tap that icon, and click "turn on". From this point on, unless you use bluetooth, the wireless button should work for wifi.
I would give the device some air, put it next to a small fan (with the sync port facing it if u think its wet), but dont take it apart unless u have to. The way parts are packed into mobile devices, leaves little room for water, if it was just rain. If it was submersed in water, it would find a way in, but in rain, i doubt a lethal amount got in if the device still works fine.
After making sure you got any water out from the sync port, try the ac adapter.
Is the power light's led responding to the charger? (Orange or green light, or even flashing would show acknowledgment)
Or you can try a sync, but the ac adapter should be more precise at telling.
If you still arent getting a sync or an ac charger to work, then something happened to your sync port and it may need to be replaced.