How do I rid my yard of this plant?
This invasive plant is EVERYWHERE in my yard, as is late honeysuckle. The honeysuckle never flowers, however. One of these plants has orange roots and causes a severe rash (worse than poison ivy) once it comes in contact with your skin.
How can I eliminate these from my gardens/beds without harming my hostas and bee balms?
The vines in both of your pictures are oriental bittersweet, Celatrus orbiculatus.
I do not see any honeysuckle at all. The leaves of the oriental bittersweet are orange. This plant does not typically cause a rash, but idividual people could be allergic to any plant. Check for poison ivy in the same area as it often grows in the same places as oriental bittersweet. Control of bittersweet can be done by hand pulling or the use of herbicides. See fact sheet on oriental bittersweet below.