The LSHT is a good place to get some endurance training, but frankly your biggest challenge on the JMT may be elevation change. That's something you can't get much of here in Texas, unfortunately. I spent several days out in the Guadalupe Mts to strengthen my legs, but even there you'll only get up to 8500 ft. On the JMT, you'll hit 10 (or so) mountain passes in excess of 10,000 ft.
Thru-hiking the JMT does require a permit and they can be hard to obtain unless you apply for one ahead of time. They do have a very limited number of "walk-in" permits that you can wait for, but a lot of people do exactly that so your chances are limited. I lucked out. I went to Yosemite to do some hiking and was prepared to wait a few days in hopes of getting a "walk-in" permit (I had applied for a permit several times months in advance, but was denied). In any case, on a whim, I decided to stop in the Wilderness Office at 4pm in the afternoon and they told me they had a spot open up if I wanted it. The catch was it was a "pass-thru" permit (meaning I had to pass-thru (and not stop) in Little Yosemite Valley) and I had to leave THAT DAY. I had already been hiking all day in Yosemite and my legs were shot, but I snagged that permit and got onto the JMT because it was too good to pass up. I got on the JMT at 5pm, but it was July so I had several hours of daylight left. The hike out of Yosemite is beautiful as it takes you past several famous waterfalls, but you also have to deal with several thousand feet of elevation gain.
Getting a permit by hiking SOBO from Yosemite is a lot easier than trying to get a permit hiking NOBO from Whitney. You have to win the lottery to get a permit in that direction. I do want to be clear on one point, though -- the permit you get for the JMT is good for the entire trail. You do not need individual permits for Yosemite, Kings Canyon, Sequoia, etc. Do NOT attempt to get on the JMT without a permit, though, as they do have rangers out there that check for them. I had to show my permit 2 or 3 times.
If you're not a purist, there is another option. You can get a PCT permit, which is a lot easier. Most (but not all) of the JMT is on the PCT, so you can hike most of the JMT with a PCT permit. You are just not permitted to do certain parts, such as Mt. Whitney, parts of Yosemite, etc.
Anyways, I'm just providing you with info. At the end of processing it all, don't let anything deter you. GO!!! You will be soooo glad that you did. I'm not sure what your hiking experience or style is, but I'll be happy to share with you how I did it (e.g., resupply, pace, etc).
