According to your site, the discharge current is 300mAh, i believe the Axim requires something like ~500mAh to charge. USB Cannot supply enough power to charge the Axim while it is powered on, while it is off it should be charging it just fine, I wouldnt see why it wouldn't be a stable charge when the axim is off.
http://www.aximsite.com/boards/showthread.php?t=98285
Instead of USB, i created something that was just pluged into the power plug adaptor.