I don't know why they said 3rd party apps aren't supported by Verizon, because they are.

By "your" texting app, do you mean one that you made? Or a 3rd party one you got from the market?

If it's a 3rd party one, simply download it, and go into the stock texting application, and disable everything. Notifications, sounds, everything. This will make it so you only get one notification for the 3rd party texting app you use, and makes it so the 2 don't counteract each other.