If I were you this is what I would do though I am not speaking from experience. So if someone out there has a better solution, by all means follow their recommendation.

The first option is to pot up the plants and bring the pots with you when you move. This is risky as some states have very strict regulations bringing soil in from another state. Of course you can use potting soil but it depends on who inspects your stuff. You can just take plants in pots and soil with you and hope noone would take notice.

The second option is to store your tubers in damp peat moss in plastic bags in the vegetable compartment of your refrigerator. When you move, just take the bags of plants with you. At your new home, you can plant them as soon as possible or when the soil is workable if you are moving to a cold state.

Since you know where you are moving, check now to see if you will be allowed to bring potted plants with you, chances are good that you will. Your roots will do better in potting mix for several months than in your refrigerator. Al

We sold our last house 6 months before we took possession of our current house, were in a rental for 6 months while building. I only brought 1 peony with me, the ground was still pretty frozen when I dug and divided it in March at our old place. It survived the rest of our crazy Denver winter/spring and the summer in a 3 gallon black plastic pot, didn't even water it very regularly, and I planted it at our new place in October when we moved in. It's been divided since and has survived the treatment with no complaints. Just make sure you put it in a pot big enough so the roots have room to grow--if the roots fit in a 1 gallon container plant it in a 3 gallon one. If you're moving to a cold zone I'd protect the pots with mulch, even dig them into the ground if that's possible. If not, a bale of straw is typically less than $10 and would probably protect several pots. Protect the pots with some shade in the summer so they don't bake if you're moving to a warm zone.
Whether you plant your peonies in their permanent location in the spring or fall is pretty much personal preference. I do absolutely believe you get better success dividing peonies in the fall but you'll be basically planting a potted plant--disturb the roots as little as possible and plant them when it's most convenient. Just make sure the eyes are at a good level for your zone and they have good draining soil. Good luck with your move, you'll be so glad you took the trouble of moving your peonies with you when they start blooming!!!

Thanks so much for your help! I think I will pot them in potting soil in large pots and move them in my car (Indiana to North Caroline) - sounds like they will have a good chance!