You should probably not let them get mowed down....you would be missing the beauty!Some peonies bloom the first year depending on the division size and food storage. Try putting a little fencing around them until they bloom. Most tranplanting is done in the fall after they have gone dormant, unless they had been in pots over the winter. I would wait to move these. Just protect them.

Is it okay to move them now, if they are still dormant?

I have a beautiful single white peony, but it's planted in the crook between the house and the fence, where no one can enjoy it but me! I would like for it to live in a slightly raised bed, mixed border design, in similar light conditions across the yard.

You should be able to move a dormant peony in the spring. I have ruthlessly moved herbaceous peonies at any time of year the ground can be dug and haven't lost one yet.

A small abused peony may not have much in the way of reserves though. Make sure transplanted peonies get lots of TLC.