You might also want to check with your local ANG (Air National Guard) and/or AFRES (Air Force Reserve) flying units that have "big iron" like C-130, C-5, C-17, KC-135, KC-10 aircraft.
I can virtually guarantee you will have better luck securing seats through an ANG / AFRES unit than with an active duty AMC (Air Mobility Command) base like McGuire (NJ), Charleston (SC), Travis (CA), McChord (WA), Andrews (D.C). With Andrews, they have an AFRES unit with C-141 aircraft (last time I checked anyways back in 2004 or so).
You need to look up their Space A number, and if they do not have one, contact Operations to inquire on flights and general information.
Many of the Space A flights for the ANG/AFRES are roundrip in nature. They fly out, then return to base (typically a 3-7 day trip), so your chances of getting a seat on the return mission is typically quite good (get to be friends with the Loadmaster, they will typically look out for you on the return and "bump" others over you guys (even if they have higher priority; at least that was my experience...). Remember, most of the crew on these ANG / AFRES flights also work full-time for a civilian company (airlines typically)
so these "out and back" missions are quite common and normally run Wednesday through Sunday or Thursday through Sunday.
Yes, I have traveled Space A numerous times:
Charleston AFB SC to Minneapolis (round trip)
Minneapolis-St. Paul IAP (AFRES / ANG) to Patrick AFB FL (round trip),
Minneapolis-St. Paul IAP to March AFB (CA) (round trip)
Minneapolis-St. Paul IAP to Kelly AFB (TX) (round trip)
Minneapolis-St. Paul IAP to Charleston AFB (round trip)
Minneapolis-St. Paul IAP to Maxwell AFB (AL) (round trip)
The best one was to Patrick AFB. Was able to rent on-base billeting in a family unit which at the time was directly on Cocoa Beach. Shared it with a retiree. I rented a car and we did Disney World.
As others have stated, be sure to have a backup contigency plan "just in case".
Best of luck!