Well generally it is better to move roses in their dormancy period, which would mean waiting for the autumn or in the spring. Waiting for the spring probably would be better, since in the winter it does not really matter if it gets shade or not.

If the plant is really weak, it might make sense to pot it and try to nurse it to better health before the autumn dormancy (which probably means that you have about a month of growing/fertilizing season left) or get the pot to over winter in a garage.

In general though sometimes it is not a shade that is a problem, but the roses itself are slow to start - I have a few in full sun, that have 3 leaves and that's about all. Sometimes leaving a plant alone is a best option, especially if the leaves look healthy.

A bit off the mark of your question, but according to another poster in zone 5 Michigan, most Austin roses are not cane-hardy there, but will be injured down to the ground or winter protection in a normal winter. This means that the plants should be set with the graft swelling below grade, several inches below where it was set in the usual nursery pot. If the graft freezes, you have only a rootstock plant left rather than the variety you purchased.

If you planted too shallow, I'd recommend replanting all of them now, since roses from pots (if they were from pots) transplant more easily during the first few months than later. Otherwise you will need to maintain deep winter protection over the crowns for the life of the plants. Forgive me if you already knew about this.

If you transplant during summer, temporary shade may be needed to alleviate wilting. Lawn chairs are good.