Ask Mr. Smarty Plants

You have a real problem, and no mistake. These are all native vines, all pretty aggressive and all are going to be difficult to remove.

Smilax bona-nox (saw greenbrier) - difficult to eradicate because of its underground rhizomes, making it very resistant to herbicides.

Gelsemium sempervirens (evening trumpetflower) - also called yellow jasmine. All parts are poisonous.

Campsis radicans (trumpet creeper) - also sometimes called "cow-itch vine"

Getting undesirable vines out of desirable plants is a huge challenge. You don't dare spray a herbicide because it would take down the azaleas, probably faster than the vines. If you try to pull out the vines, it will result in damaging the azaleas, pulling off leaves and flowers. Likely your only chance is perseverence and stubborness. With gloves on, begin to clip off the vines wherever you can get at them. Clip them into small enough pieces that they can be unwound without unnecessary violence to the plants you are trying to save. Keep working back to close to the soil, and then try to work the root or rhizome out of the ground, again without damaging azalea roots. You can, of course, start by cutting every vine stalk as close to the root as you can get, and then waiting for a while until the vines without roots start to turn brown, making them a little easier to find, identify and remove. You can also try painting that cut root surface with an herbicide, using a disposable foam brush. If and when you get the bed cleaned out, you will have to be constantly vigilant. All of these vines are very persistent and they will come back from underground roots, rhizomes and seeds. It's much easier to pull them out when they're very young than after they grow up-but you already know that. And dispose of the cuttings in such a way that seeds will not have a chance to resprout.