Thanks everyone for your messages. Zuzu that was hilarious! You have obviously had lots of experience with gophers. We tried the exhaust thing, too, Porkpal, using our generator. They could be voles, Arlene, as some of the new azaleas have bark stripped on stems and leaf die back.
I have decided to re-dig the roses, put them in cages since that bed will be a dedicated rose garden and I won't be digging into the cages planting other things. Also, will use the new trap for the perennial bed and if we get one, for the pure pleasure of putting it out for good!
I have learned from all of your messages....so far have only grown Knockouts other than Julia (which I am afraid is a goner) and I didn't know a rose had no tap root. So, the cages I am building out of hardware cloth are "squat" like recommended (short & fat) so if that rodent gets in that bed and gets some of the roots that have outgrown the cages, maybe it won't kill the whole rose. Thanks again!