I can't be certain, but I would first say to check whether or not your backup battery is good or not. If it's bad, it should at least be easier to replace than the motherboard. [See
http://www.aximsite.com/boards/x50-h...-low-batt.html for more (related?) info.]
To answer your other question for the function of the backup battery: yes, it is supposed to provide juice after the main battery drains or when you switch out the main battery... BUT only to the Axim's standby mode (which AFAIK is the same as when you turn the Axim 'off'). The Axim's RAM is still being powered by the backup battery, but due to its smaller capacity, I think the backup battery is only rated to provide 30 minutes of power to the RAM.
Either way, I think it's bad practice to pull out your main battery when you still have the Axim on. If I remember correctly, whenever I've tried that I also got a memory error when turning the Axim back on.
In short, I would recommend:
-Only replacing the main battery when you have the Axim turned off.
-If you are still getting memory errors after this, try a hard reset.
-If this still doesn't help you, I would investigate seeing whether or not you can replace your backup battery before giving up on your Axim.