I've never heard of a caterpillar being poisonous but then I don't know if there are any that could be either.
Whatever the case I would not take a chance because some do sting and the sting is very painful.
I know I use to get stung by the asp and I think they are way worse then a wasp or bee sting.

They sometimes fall out of trees and land on you.
I hadn't seen any for a long time then when we moved here I started seeing them again.

There are some pretty nasty caterpillars out there! The "puss caterpillar" comes to mind and it supposedly delivers a really painful sting. Here is a link http://insects.tamu.edu/fieldguide/cimg256.html There are 4 other poisonous caterpillars listed which are common to Texas.

Sounds like webworms to me. They will eventually become moths I think.

Pete

Blast, they are webworms. And they're on EVERYTHING. My neighbor's catalpa is just covered in them.

Should I be concerned for my bedding plants, or will the caterpillars stick to the trees?

They are pollinators FYI.

Spraying bt will control them. It makes them stop eating and die and isn't harmful to humans - just caterpillars. Just be careful about where you spray it in case you do garden for butterflies. Catalpa trees are famous for getting those caterpillars on them.