Hi Pat Williams
Installing and uninstalling printers can be a complete pain as it needs to be done correctly and can be time consuming.
If you do not remove all of the previous printer software or you have the printer connected to the PC before being prompted to then the printer will not install correctly, Dell PCs can also be difficult with other branded drivers at the best of times.
My suggestion is that you remove all printer software from your computer and use the HP uninstaller if there is one available, shutdown the PC, disconnect the USB cable, reboot, install the printer software and only connect the printers USB cable when you receive an onscreen message telling you to do so, hopefully this will resolve the issue but if not try another USB port or data cable.
I haven't been able to find the original disk anywhere
There may be a Driver Reset Tool application installed on your system, if you need to reinstall a driver when you boot up try pressing Ctrl + F11, this may not be present on PCs that were provided with an actual restore CD, other options;
Click Start, All Programs, Dell Accessories, Driver Reset Tool or Click Start, Run type in C:\dell\utilities\driver reset then click OK