This is why I despise 'Dr. Huey' as a rootstock. You can be guaranteed that at some point in the life of the rose, 'Dr. Huey' will send up suckers that you will never, ever be able to stop. This is just what it does. The joke around here is that it is everybody's favorite rose because in May, there is no other rose you will see more blooming plants of than 'Dr. Huey'.

Dr. Huey IS the rootstock. Once Dr. Huey starts to sucker, you have to dig down under the soil, find the source of the sucker and tear off the sucker at the base. Otherwise, you will wind up with plants of Dr. Huey and not your original rose. Which is what you have now. Just cutting down the suckers does not help, you are just pruning Huey. If you can't get down to tear off the suckers, and if there are too many now, your original grafted plant is a goner, as Dr. Huey is very vigorous. Unless you can save cuttings of your grafted plant, you will have to dig up the Dr. Hueys and start over.

You can try to avoid the suckering by burying the graft union underground, but as Paul said, in some places Dr. Huey is just bound to take over.