All good questions, macseann. Here's the short answers:
What was the trigger event that shifted Texas away from annexation efforts with the US and towards a British negotiated peace/recognition of independence with Mexico?
There really isn't a trigger event, since support for annexation was almost universal in Texas. But the Texas Secretary of State, Ashbel Smith was working on a treaty through the British, known as the Cuevas-Smith Treaty, that would have resolved the border question peacefully. You can find it on-line in a few places (I have it in a book) it's very short, the language is very broad and made a great place to start for an AU since Texas already had a navy in real life.
Quite a few towns and counties are named after American heroes, but I wonder if that would have started to taper off after the Mexican War, or would continued immigration from the US still have brought with it attachment to the mother country?
I try not to change place names. Besides, it's easy to change the backstory for a place name. My ancestors settled around Thorndale it what became Lee County. I've been through Normangee a few times.
3) Where are your primary naval and air bases at this point (Depression era) in Texas? I've been reading up on some of the air bases in Texas for a history podcast I'm working on, and it would seem that some of the names (though not the locations) would become problematic in your timeline. Most of the extant airfields in the 1920s were named after Army aviators who were killed in crashes, and many of them, such as Love, Dodd, Ellington, etc got their names from aviators who died in places other than in Texas.
The primary naval base is the Galveston Navy Yard, with a submarine annex in Texas City. Corpus Christi and Houston will get added during WWII. Airfield names don't change so they're easy to find; I just change the backstory for whom they're named. If you want a good on-line reference for some of the pre-WWII airfields in Texas, look here:
http://www.niehorster.orbat.com/013_usa ... coast.html