Do I just need to pull the old hard drives and slave them into the new system to at least pull off my files? Will that work?
Yup.
When your new PC arrives, lets assume it has just one hard drive. Remove the hard drive from your old computer. Set the jumper on that hard drive from "master" to "slave" - there's usually a diagram on the hard drive itself that shows the correct position. Find the wide, flat cable on the new computer that goes to your new hard drive. There may be a spare plug on it. Alternatively, it may also be connected to a CDROM drive. You can remove the plug from the CDROM if so.
Insert that plug into your old hard drive. Find a spare power connector (that's the other plug that goes to the hard drive) and plug it in.
When you next switch the computer on, you'll probably need to go into the BIOS (press F2/delete repeatedly) and aout-detect the hard drives, so that they are correctly reported to your new operating system.
You do not incidentally need to physically attach the old hard drive to the new PC. So long as it is near horizontal or near vertical, and the leads aren't strained, you can rest it on a pile of books. Just make sure the circuit board on the underside isn't shorted.
Then boot into Windows. Transferring the files from that point on is just a matter of drag-and-drop.
Having secured your files, you can almost certainly rescue your old Dell, with a full reinstallation of Windows.
