Welcome to Texas (almost). I am a USAF vet too, retired, and can't hear real well either.
Texas does recognize Missouri concealed handgun licenses, or permits, or whatever the Show-me state calls them. As long as your license is current and you are a Missouri resident you should be good to carry on it.
You can get a Texas non-resident concealed handgun license for people who reside outside of Texas, and for people who intend to make Texas their residence, but you have to take the class in Texas. I don't believe it is a big deal to then change that to a resident license once you move here, a few bucks for an address change I think, and I believe you will need a Texas DL, but I am not familiar with that process. Your CHL instructor will be able to help and I put a link at the bottom to the DPS website for CHL stuff.
You do have to shoot for the initial issue (a couple exceptions), but you can shoot revolver or semi-auto, doesn't matter. Currently the smallest caliber you can qualify with is .32, but that may change to .22 later this year.
You generally establish residency by intending to make some place your residence. You can document that by having an address there, registering to vote, establishing bank account, i.e. doing things that a normal resident would do. To make it clear, you should probably notify Missouri in some fashion that you are not longer a resident, e.g. by cancelling your voter registration there, if there's income tax in MO then there might be a form to file with your final income tax return. (That's important - Indiana used to try to tax people who moved away and didn't make any formal declaration of changing their residency, a fellow USAF officer once found he had a warrant for his arrest in Indiana for failure to pay income tax for a few years after he moved to Texas. Don't know if Missouri is that greedy or not, but wouldn't surprise me).
For a resident license, when I first moved here, I had to have residency for six months, but I believe that has long been done away with. But in any case you can get a non-resident one through the same process. For a resident license I do believe you have to have a Texas drivers license, and you are supposed to get one of those within 30 days of making Texas your residence.
There are a number of CHL instructors on this board, I am sure they can help more with specifics, and with any luck maybe some of them are close to where you intend to reside.
Judging by the comments on this board, licenses are moving pretty fast once all the correct paperwork is turned in. DPS has 60 days to get your license to you once they receive all the required paperwork (and assuming you are eligible). If your package is missing something, the clock resets until you get it to them. Seems like it's been going faster than that tho. There was a problem with backlog a couple years ago, but seems to be fixed now.
Here is the Texas DPS page on CHLs, lots of good stuff here:
http://dps.texas.gov/rsd/chl/index.htm" onclick="window.open(this.href);return false;
Again, welcome!