Haven't moved either of my diabetics but I have moved cats....alot!
What I have found that worked the best and seemed to stress them out the least was to keep them in a single room at the old place, usually the bedrom which I would leave that bed set up until the last moment, and while the bed was stripped for the move I would leave an unwahsed blanket of comforter on it, so they had all the smells of home. This also keeps anyone from scooting out the door while boxes and other stuff gets moved.
If it is a short move I would move all the non-essential stuff first and get that all set up in the new place. Then the day of the move, put them in their room, pack up everything but what is in there room, move that, get all the big pieces where they are going to go, get litter boxes, their dishes, condos, trees etc set up, then go back pop them in their carriers load them up and put the blanket that they had in their room over the carriers. Then move the furniture from their room get it set up and bring the cats over last. Put them back in the now new room with the familiar stuff from the old house and let them out in there to explore once they are relaxed in their new room, open that door and let them explore the new digs.
I also usually set up several Feliway plug ins in the new place to help with the transition to the new place and so far, no problems with anybody getting anymore stressed out than a trip to the vet's.
Mel, Maxwell, Musette & The Fur Gang