For the short move, I would probably wait until you’ve moved to set up the new tanks. However, it’s a great situation since you plan to have a 1 month overlap, it’s close, and you can move the aquariums at your leisure. It might even be useful to set up the larger tank in a very basic way, get it cycled, and put a few fish in there to keep it cycled. Then when you move, you can store all of the fish from your current tank in that one, while you set up the current tank in the new place without any stress, and move the fish whenever it’s ready.
In regards to the long move, 2 years ago I moved about 1850 miles. Prior to the move, I broke down all but 3 tanks (a 125g, 68g, and 20g) and either traded those fish in to the local fish store or consolidated them into the remaining tanks. I had the moving company build crates for the tanks and move them, but moved the fish, plants, and filters myself.
I underestimated how long it would take to break down and store all of the fish and materials for the trip, and it took me all night to get ready. I waited until the last night, because I didn’t want the fish in bags any longer than absolutely necessary. So, I was actually just vacuuming out the last tank when the moving van arrived. I highly recommend a wet/dry vac. I don’t think I would have finished in time without it.
The fish went into breather bags with a few drops of Kordon Fish Protector (whatever the directions were), and were then stored in coolers, separated by cardboard and shredded newspaper. I think I wrapped the plants in wet paper towels and bagged them, but I didn’t give them as much attention, because I was running out of time. I put the sponges from the filters into bags with just a small amount of water, just enough to keep everything damp. Hardscape and substrate I wanted to keep went into 5 gallon buckets and were shipped with the tanks. Lights came with me, and I still lost 2 of them for a month, lol!
Then we drove for 4 days. The coolers with fish were kept cracked slightly open for air exchange. They had to travel inside the car and be carried into the hotels every night to protect them from the summer heat, sometimes up flights of stairs. For more fun, we were also traveling with 2 dogs and a snake.
Upon arrival, we went to the new house, and I set up the fish and surviving plants in large Rubbermaid tubs in the basement with makeshift lids, and sponge filters. All of the low-tech tank plants survived. Most of the high-tech tank plants did not. Most of the fish survived (around 70). I lost 3 large fish—a pleco, a catfish, and an acara.

It’s not terrible statistics, but I really hated it for them. The spiny fins burst the bags, so I don’t recommend breather bags for those types of fish. Get them a bucket and battery powered air stone instead, I think, or trade them in before moving. I did have 3 other large catfish and another pleco that arrived just fine, but I wouldn’t do it that way with them again.
The next day, after the movers left, I started setting up tanks. They cycled quickly (within the week), so I think quite a lot of the beneficial bacteria survived in the filter sponges.
There you have it, good and bad. Hope it helps when the time comes!