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Run MSCONFIG and go to the startup tab and make sure nothing strange is being loaded. You can uncheck startup items to troubleshoot. Recheck them one at a time, you may be able to isolate th cause of this. From there you'll need to figure out how to handle the situation if something is found.
You can use system restore option in Win ME to try and go back to a date when the PC was working.
Any recent additions may be causing this problem so uninstall any new software/hardware.
By the way I've been using Win ME since it came out and haven't found it to be any more or less unstable than any OS. One thing I've noticed is, there hasn't been no where near the amount of security bulletins, fixes, updates and patches as Win 2000.
Before you go out and spend the money to get a new OS I'd use the system recovery that came with the PC. And that's only if all else fails.
This problem is something that can be fixed. To start over shouldbe a last resort.
The fact that it's a Dell menas you have a few more options. Like calling tech support, a good warranty policy and you can get the drivers and all files needed for your system.