You can, but it would be better to do it in the fall. I don't know where exactly in GA you are, but here it's been very hot and dry. You'll have to water the rose almost every day and it's sure to have some kind of transplant shock from being moved.

If you do, I'd water it well for a few days before you move it.

Only if you can dig it up losing abolutely no soil from the root ball. If any soil falls away it will be like planting a new bareroot, and in the heat we have at this time of the year, it would be almost certain death. Your water restrictions could be a problem also as you would have to water deeply and often until it got
re-established again.
If you decide to chance it, cut back the top growth so the roots don't have much to support. It would probably die back anyway.

Yes it sounds scary.. will reserve its spot and move in Fall. Its just not doing as well in its current spot and I know it can do so much better!
Thanks.