There is no need to pot anything. You can start layering the blackberries right now. Pin down some growing tips and then leave them in peace. The raspberry canes just need digging when you are ready to replant. The best place to store them over winter is right where they are now. When you are in your new place all you need to do is wait until the ground defrosts then nip over to your parents to dig your plants. There is no need to wait until after last frost. These things are bone hardy. Just so long as you can dig them out and then make planting holes.

Interesting, I didn't know I could dig them up like that after the roots had spread. I assumed I needed to replant the entire root ball in a way I could only get by rooting into a small pot!

They won’t grow a huge root mass between now and next spring. And they really don’t care even if you do tear a few roots when you dig them up. If you prefer you can pin the tips down into pots of soil rather than just into the earth To make moving them later easier. However, I don’t know your climate so I can’t say if the pots would freeze over winter damaging the roots.