First off, have you tried the Dell hidden system restore which is available on newer Dells. Depending on your computers age you may find it doesn't install a lot of the extra programs.
To restore your computer press Ctrl-F11 when you see the blue Dell bar at the top of your screen, after the bios
but before Windows load.
As to your other question:
Yes, it MAY work.
It depends on the other disk you are using.
Disks from major manufacturers like Dell, HP, etc. check for the product key which is actually in the bios
. If you use a generic XP oem disk like you would buy from a retailer of OEM products you can just reformat and reinstall and put in the key found on the computer. You will have to activate your XP if you do it with any disk other than a Dell.
You cannot use a retail version disk of Windows XP to do your re-install. They use different keys.
You MAY be able to use a disk from say, a Gateway, but often that won't work because there is software on major manufacturer disks that accesses the above mentioned bios
key and often it will see its not the right manufacturer and abort the install (this keeps changing so I can't tell you which manufacturers will/willnot work.)
That being said here's your best option.
Find someone with a Dell disk that matches what you have (XP Home or Pro).
If you find one that already has SP2
even better. All Dell XP disks will work on any Dell as long as you have the matched the Home/Pro that was originally installed and is on the product key on the machine and then you will not have to activate it.
Trust me on this. I have a computer shop and have done hundreds of installs and I guarantee I have given you accurate info.