It sounds as if the root stock to which your roses are grafted is taking over. The red rose you describe is probably Dr Huey a vigorous once-bloomer often used as a root stock. If you want the original rose to survive try to tear off the shoots on which you see the red rose; the leaves may also be clearly different on the shoots which have not yet bloomed. Ripping them off is better than cutting as they are less likely to grow right back.
