I would leave the locust for now.

I find the best way to work on honeysuckle is to concentrate my efforts in the spring: during the spring, the ground is damp and honeysuckle is easy to find since it is evergreen. Grab a piece and gently pull it - sometimes you can pull a large part of the network at one time. I take a black plastic trash bag with me and just stuff it all in there. You can't leave it on the ground or it will resprout.

The more you do this, the more effective you'll learn to pull. And of course, take your pruners and immediately cut off anything that is in the trees.

Pulling the vines as suggested above will help keep the infestation downs, but they will sprout vigorously from the roots.

After doing some 'damage control' during the growing season, you may be able to get lasting control by using herbicide (glyphosate/roundup) during the winter, when (hopefully) desirable neighboring species are dormant. I have had good success spraying the green honeysuckle foliage during November-December in Missouri, though there may not be the same window of opportunity in Georgia.

Black locust is a lesser problem, espeically since yours are so small. Since they are small, you can probably kill them pretty easily by cutting and treating stumps with a brush killer. But I recommend that you ask around to find if this species is a 'problem' in your area. I thought this species was more of a problem (due to invasiveness) in the Northern states. Although not the most desirable native tree, the beautiful and fragrant blooms are very appealing.