Do you know the names of the roses?
And when you take the photos, try to get some 'normal' growth in as well.
For how many inches are the thorns soft?

Oh Anne, I'm just sick to my stomach right now! I just went to get a close look at all of my roses. The only part that is affected on any rose is the new growth that has grown since I deadheaded them, so only about 2-4 inches right now. The one that looks the worst is Velvet Fragrance, which was also closest to where I brushed the roundup on the weeds. I have one rose, chrysler imperial, to the right of velvet fragrance. The new growth appears normal. I just planted it so it wasn't there when roundup was sprayed. To the left of vf I have memorial day (I can't tell if it is affected because it doesn't have any new growth on it yet), then double delight, golden celebration, frontier twirl, tahitian sunset, all of which have very soft thorns on the new growth and the leaves are curled. The leaves are normal shape and size. I also noticed that some of the buds are completly bent over, as if they are in severe need of water, but they aren't limp, in fact, if I straightened it the stem would probably break. They have been watered regularly and are not dry.

All of these are by our driveway and I know that my husband sprays roundup in the cracks where grass grows up. I've never thought anything of it since I had no idea it could cause trouble. Is there something in particular to look for to determine if it is RRD or roundup damage? How long does it take for roundup damage to occur? The roundup was sprayed the same day I pruned all of my roses in March. All of them grew like crazy and were very healthy with lots of flowers.

I suggest that you and your husband stop using Roundup around your roses. People have gardened for more than two thousand years without using that herbicide. Roundup and fertilizer cause far more damage to roses than weeds and growing in native, unamended soil.

In total agreement with Cass.
My gut feeling is roundup. And roses do recover from it, if they don't die instanteously.
Especially if some of the weeds y'all were treating were woody, long lived , your older roses could have shared root grafts with the weeds.
Also, your part of the world has had a lot hotter temperatures that we in the east. There is a danger that chemical can become volatile. Drift of chemicals is a widespread problem as well. (And it's been studied more because there are now Genetically Modified strains of corn and soybeans that are Glyphosate resistant.)

Do I need to cut off the affected parts of the roses? How long will it take for them to recover?

My husband was mainly treating grass/weeds that were growing up in the cracks of the driveway. I have no idea if they were woody. How do roses share root grafts with the weeds? I planted all of these roses last year. Will the grafting happen that fast?

The paper published last year showed Dr. Huey to Dr. Huey root grafts happened in six months. And a paper from China showed the possibility of a Pawlonia tree and roses having the same sharing, although that one has a lot of other unstated working hypotheses.
I've had recovery in one year. A friend who had a family member spray his roses in his absence and the member used the wrong sprayer, for his roses total recovery took three years, but his were very close to dead.
By cutting off the damaged area, you are removing the potential for addition spread of the bad stuff, whatever it is, within the plant.

I agree about Roundup and fertilizer damage. At the same time I wanted to mention something I use when I feel I have to use Roundup. Bindweed is my scourge here. I cut the bottom out of coffee cans. I place one over the weed and spray at very low pressure. I leave the can there until it dries completely. I learned the hard way and dripped herbicide off the can when moving it. I tried painting it on with a brush and applying with an eyedropper, etc. This is the only way I apply Roundup these days.

Good luck! I haven't gotten any on the roses. Can't help on whether to remove foliage or not.

Assuming this is roundup damage, will my roses continue to grow like this until it is cured? What does it take to cure it? Will they bloom at all during the recovery? I'm still going to post pictures later today so hopefully that will help.