NO, you can't move a 20 gallon tank loaded!
You cannot move a 20 gallon tank full of water -- bad, bad idea! Hopefully I misunderstood, but that is a recipe for disaster. It is hard to keep from twisting a glass tank when it is being carried with water in it. That is why you should always move tanks empty. The glass does not do well in twist stresses, too easy to rupture a seal and you may not know it when it happens, it may fail later.
You have two choices.
One, bag the fish. Do not feed for 2 days, bag each fish separately, carry in a bucket. I'd use a 20 gallon bucket, or large ice chests.
Or, two, just use the ice chests, large styrofoam ice chests are fine and you can get a new one and not worry about whether it has had cleaners in it. What you want to be sure of is that the fish do not swim around in circles like inside a large bucket, they can scratch an eye that way. They only need enough water to cover fins and let them be upright during transport, you can add more water at the new place. Probably good to do that, in case the water feels different to them.
When I moved large cichlids across town, I added Amquell to the ice chest, plus a heater and airstone when I got them home, and I took the time needed to set up the tank, it takes a few hours to do it really. The filters were cleaned and the media removed to a shallow ice chest with an inch or so of tank water in it to keep media moist not submerged. Everything was fine when we got it all set up.
That 100% water change on set up gives you a head start on any ammonia bump, and if the fish are unhappy, a dose of Prime will help them feel better.