Holly,
John is right about the stuff for bringing it home. You don't need the ax and bow saw for collecting firewood

But since you asked for "Ultimate Airstream Road Trip List" that's what I gave you. I didn't mention it, but I actually carry a small Stihl chain saw along. Less work cutting up dead limbs for firewood. And I left out the water spray bottle for ironing and making cats behave! LOL!
We also take along the Canon S900 Photo Printer, not to mention a
well-stocked bar. If you want to know how we fit it all, check out
The Long LONG Trailer in my signature. And we aren't even full-timing yet (soon). Our two cats are declawed, so we don't worry about them in the trailer.
We put paper towels between the dishes. Someone on the 'net told us to put glasses in socks. Works great! The pots and pans nest in each other in the frying pan which is on the pizza pan on an oven rack. I bungie the oven handle to the cooktop. For the bar, every other bottle comes out, I put in a strip of that thin packing foam 12" wide and put them back so the foam weaves between them.
When we get the final TV and stuff, I'll get some TV Grips, but for now I put it between the bed and wall with pillow around it while we're rolling. It's also hard to beat the rubber cabinet/shelf liner that keeps stuff from moving. Museum clay is good to hold vases, pictures, etc in place.
For your trip home...
I'd definitely make sure they give you the tongue jack foot (and the stabilizer jack cranks as well as the manual crank for the tongue jack even if it s electric), and have at least a coupla chocks (two 18" 2X4's nailed together make a good chock). I like to have the board for the jack foot... ground may be soft. It's also good for placing a bottle jack on if you have to change a tire on the trailer. If you're going to fire up the frig, have some leveling blocks also. Pieces of 2X8 lumber with an angle cut on the end, the yellow Camping World legos, or the orange Lynx Levelers will do. You don't want to run the frig off-level.
If you're going to camp in it on the way home, sooner or later you're going to need a coupla fresh water hoses, a coupla sewer hoses and the attachments (Airstream dump valves are different from other RVs so you may need an adapter), and you may need the 30A extension cord and/or the 15A/30A adapter, so may as well have them to start. Ask the previous owners if they're providing them.
I personally wouldn't camp in it unless they've added a LP detector, or I'd turned the bottles on and went over every connection myself with soapy water. Seriously. Ask if the bottles have OPD valves. If they don't, you may not be able to get them filled in some areas and may have additional expense getting them retrofitted. If they've been sitting empty, I'd have 'em purged and refilled before departing. Personally, I'd also tow it with a full fresh water tank so I'd know if it leaks before I got it home. You can also test the water pump if you fill it at their site.
If you're going to be camping with an older furnace, refrigerator, or hot water heater you're not familiar with, I'd find a battery powered CO detector for the trailer. Seriously.
Look at the condition of the tires carefully. Make sure there's a spare and it's in good condition, and ALL are correctly inflated. By the time you realize you have a flat, the tire may be too far gone to repair. Check the pressure in them yourself. Hopefully, they'll have proof of the last time the wheel bearings were packed and the brakes were checked. Hook it up and check the brakes yourself before making final payment and departing. Check all the running lights, turn signals and brake lights.
John made a good point about the lug wrench too. Make sure you have one that fits, or get one before you travel far.
I'd also bring a bottle of distilled water. If all works correctly, you're going to be charging the batteries as you drive on the trip home, so you want to start with full cells. Personally, I'd bring a battery tester along and negoitate the price down if they aren't full and fully charged according to specific gravity.
That's all I can think of off the top of my head.
FWIW, I graduated from high school in 1969! That was a long time ago! I'd have loved to have had the time to restore an A/S from this year!
Hope this helps! We're real happy for ya!