I am no expert, but I think that particular rose bush was grafted to a rootstock of a once a year rambler. The rambler took over and the original rose died. Are the leaves now different than when it did not grow those shooters? I had experienced with my roses that were pink turning red, it was grafted to Dr. Huey root stock and the root stock grew branches and I did not noticed, eventually it grew stronger and the grafted original rose died and so I had a Dr. Huey rose. I noticed the form of the leaves also changed---they are the leaves of Dr. Huey. If that is the case, there is no fixing, unless that same bush still has parts that has different leaves. Then you can cut off and take it. off the growth from the rootstock---
