While a few people have reported problems with the Dell backup utility, I've used it successfully a few times. In Start, Programs, Data Backup - check the box "Back up PIM database only" and select SD Card from the dropdown list next to "Backup to:". I would also give it a different name from default (some like PIM-12Mar). If you've got room on the card, run backup again and select "Back up all data," giving it a different name. Most likely if it is a software problem, you won't be able to restore the full database but you won't know until after the hard reset.
Go ahead and back up, but before the hard reset check
Chris De Herrera's site. There are a lot of potential problems with ActiveSync that may have nothing to do with your Axim (the same problems occur on other PPCs as well).
If none of the solutions there are useful, do the hard reset and try syncing before restoring. If it still doesn't work, Tech support may be right but at least you'll have the data for use on another Axim (note, the backups will not work with other brands and I'm not absolutely sure it will work on the other Axim model but I suspect the PIM-only file will).