There's absolutely no reason to be afraid of being close to the border. You're reading too much 'glorified' news stories! :)
How about 1 month in each place (FL, TX, AZ) then you'd have a chance to see both children? You'll also get a feel of each place and it will be easier to choose the next year. If you're 'active' folks AZ is the best for beautiful hiking along with geocaching, golf, 4-wheeling, tennis, pickleball (the new craze) and many other activities. We absolutely love Arizona best of all three. The last time in FL we planned to stay 6 months. After 2 weeks we were out of there (bugs, humidity, rain, cold, crowds, traffic). The Rio Grande Valley of Texas was the same way but add the constant wind. Arizona has no bugs (in winter), no humidity, mild temps in the 70's for the most part with a few cold days but quickly warms up, and the spring winds don't start until March/April and it's not every day.
Have fun planning!