What can be added to veggie burgers to increase texture?
Instead of adding ingredients as an answer, how about tackling the TVP texture?

Try many-hour soaking first off. Drain and let steam in it's own 'juice' in a tight-lid pan (not too big) on low for just a few minutes. Done at your desired chewiness. 

This has the effect of giving off some of the soaked up liquid that would otherwise wick off during burger frying.

Can add in your seasoning while cooling. Binders best added when room temp.

Now, for texture and sogginess prevention, my fav vegburger addition is cooked brown rice. A little goes a long way.
Also, terrific crunchy chewiness with fine chopped raw broccoli.
Raw sunflower seeds add meaty-ness if burgers are fried several minutes.
If quicker fried then maybe soak them first. Pumkin seeds too.