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.
I think that Kelly is trying to say that VZW doesn't "provide" support for them, not that they don't work on the phone.
The post above is correct. You just need to disable all of the notifications of the stock app and turn them on for the 3rd party app. You don't really "disabled" the stock app, you just keep it from notifying you of anything.