Not the ideal time, but you have no choice. They should make it. And the canes that fruited on top, will fruit in the summer on the lower 2/3 's of the plant. Once that happens, cut them down right at ground level. New primocanes will be growing too at that time.
Most advice I have heard is to transplant raspberries before they bud in the spring. That is the ideal time.

It probably is a good idea to cut the upper third off. Any fruiting parts should be removed. The plant will try and produce the fruit instead of concentrating on new root growth. So any that are left, cut cane just below all fruting sections.

I have transplanted everbearing raspberries here in Madison, WI, in the spring, and also in the fall, with success. Just water them in and mulch them, and they should be fine.

I transplanted all my raspberry plants last fall and all survived but did suffer extensive rabbit damage during the long MN winter as I did not have a fence around the patch. I will also be transplanting several plants this weekend as a few family members are now interested in beginning their own patch.

Prune them back before you start digging and they will be fine. Fall bearing are fruiting now, I would think that is more likely what these are.

Thanks for all this superb advice. Wasn't sure how tough raspberries are.
YES, I do have those pesky rabbits but not many.
(neighbors hunt them for food so they should not be a major problem.) Judy