Not sure what's in your tank, but I would think you could use your current tank water in the bags and cooler you use to transport the critters, and dump the rest of it. Prior to that, at your new place, have a good sized tuffy trash can filled with RODI water with salt mixed in and a heater to bring the water up to temp. Then when you bring the empty 20 gal and cooler full of critters, you can fill up the 20 with the new water and begin the acclimation process to the new water. It will be the same amount of stress (to the critters) as moving to any new tank. But the old water (in the bags/cooler) will be pretty nasty.
If it was me, I'd build a new system that I could grow into (like a 75 gal) and get that all setup and established, then move everything into the new system. But thinking back to college days, that might be easier after getting a few paychecks!