Looks to me like those buds are new basal shoots coming from the graft. Maybe somebody can confirm.

I plant my roses with the bud union about 5cm below the surface so I normally only see the shoots once they surface. If a shoot has come from the root stock the foilage is normally different and then I go down and tear it off - never cut.

Yes, the swelling itself is all scion, or the desired variety. Rootstock shoots come from below the swelling.

You should have a pretty "new" plant from your newly forming canes.

I urge you in the future to bury the graft at least several inches below ground--to protect it during the winter. In fact, some northern gardeners bury the graft 4-6 inches below ground level--just to be safe. If the graft suffers winter damage, it is all over for your rose.

We'd like to see pics when your rose finally blooms. : )

Kate

You need to prune off all the dead wood and let those new basal breaks grow out. Any cane that has a brown or tan center is dead wood. Clip off a bit at a time down from the top until you find a nice clear greenish white center in the cane. From there on down the cane should be alive. Congratulations! Looks like your roses survived a hard winter!