A core charge is what would be charged to you if the seller requires your old injectors. Same as alternators, heads, brake calipers. Most places require a rebuildable used core in exchange for new parts, or they will charge you for them if not furnished.
Reman just means that they are not original parts. They were not produced by the original company, ie-navistar. They are not susceptable to the same specs, but were made to replace without any modification. Just like a reman'd engine or transmission replacement from Jasper. Most times you are better off to go with oem parts or even an upgrade. But you can also get lucky and never have a lick of trouble out of them. In my opinion, if you replace one, replace all of them. A new injector may shoot more or less fuel on your piston then the other 7 and create an off balance on your crankshaft. It woud be a rough idle at all times, and could eventually really mess up your engine. Spun bearings, connecting rod failure, even broken cranks may result.
I really wouldn't pursue injectors until you have followed up on bleeding your lines, Let it idle and open each line at the injector for 5 seconds or so. Do this one at a time. If you never see fuel coming out, leave it cracked open longer.
Good luck with everything and let us know how it works out for you. I'm off to Baltimore again!

 

I just bought a set of reman Delphi injectors from Pensacola Fuel Injection for $123 off Ebay. That included the injectors and shipping and no core.

When checking the glow plugs as mentioned above you need to disconnect the wiring harness from the glow plug.

 

Well fellas I've got some good news and some bad news. The bad news first...I worry too much. The good news is, the beast is running great again. Turned out to be nothing more than some air in the fuel lines. My tach is also working again for some odd reason. I bought a new tach sender and was trying to replace the old one. I disconnected it but could'nt get that dang nut off. I sprayed it with some WD-40 but still nuthin...So I just reconnected the old one and I'm gonna wait to get a 1" wrench. Well I started the beast and the tach was working again...

 

Sounds like a dirty connection at the sending magnet. Clean up your probes in both sides of connector, and put a dab of grease on them before mating back together. Hell of it is, you can't return electrical parts to ford. Atleast not where I'm at anyhow. And in my opinion, you might as well drop your drawers and squirt on it. It'll do just about as much good as wd-40!