I moved (in a small car) from northern CA to Colorado in January. It wasn’t nearly as far as your move, but I think I can answer some of your questions, in terms of my experiences.
First, I had been preparing for the move for a long time (nearly a year). I knew I would be moving at x time, therefore, it was easier to prepare myself and my friends and family. I was visiting Colorado regularly to see my boyfriend, so I got to know the area I’d be moving to, and I checked out some neighborhoods. I decided on a neighborhood and flew out here one last time in December to find an apartment. Luckily, I found one pretty easily that was ideal. I job hunted for about two months before I moved, but the economy at the time was so bad that nobody gave me a second glance since my resume still had a California address. So I moved here with no job, but I already had an apartment to move into, in a good neighborhood, that I was happy with.
Second, my boyfriend and I made the trip together (since I didn’t drive, there wasn’t much of a question about this, actually). We planned out a road trip - so rather than driving straight back on interstate 80, we went the long way around, driving all the way through CA, AZ, NM, and back up through CO to Denver. It took us 8 days (IIRC); we stayed with friends and family while in CA and cheap motels in AZ and NM. Both of us got to go/see places we’d never been/seen before, and I got to spend time with my relatives before I moved. I think doing the move gradually like that, making a fun trip out of it, made that much of a difference in terms of stress level. It was stressful having the car full to bursting, but other than that, it was great! And, it was a good test of our relationship (since we’d been long distance for a year and a half before that, we’d never gotten to spend more than a few days together at a time). Luckily, we got along really well, and learned that we have similar travel styles. We have a lot of fun together in all kinds of situations, so it was a good trip.
I think it was scarier for my boyfriend to have me move here than it was for me; for me, it was mostly an adventure, and an escape from my parents’ terrible separation. Once I got here, and my stuff all got moved in (and mailed boxes retrieved from his parents’ house), I said WOW. For the first time in my life, I had my OWN APARTMENT. I could do whatever I wanted with it, and inside it. It was great! The stressful part came when I spent weeks of fruitless job hunting, and it took 6 weeks to even get a temp job. I kept busy by refinishing furniture I found in alleys, getting familiar with the neighborhood and the city, and spending a lot of time in the library. Eventually I did get a job, met new people, and felt really comfortable in my new place. The first couple of weeks were hard - I was 1000 miles away from almost everyone I knew! but at the same time I loved it.
My advice to you? Plan out your route ahead of time, but make room in your trip for little side trips (if there’s room in your schedule for that, and you don’t HAVE to be in TX on a certain day). There’s a lot of amazing things to see on your trip, no matter which way you go. Maybe take some time to go places or see things you’ve never seen before. Take a route that’s more interesting, rather than the fastest way. I loved seeing the Grand Canyon in January, because I could SEE it, rather than being behind tons of people all there to look at the amazing hole in the ground. If you can plan out an estimated route, maybe post here and ask dopers for suggestions of what to see in these states/areas along your path. For instance, if you go through South Dakota, be sure to see the badlands and Wall Drug if at all possible (IMHO).
Buy some books on tape/CD if you can afford it, or ask for them as gifts, so you have something other than radio/music to keep you company. Do you know anyone who lives along the way? I bet a lot of dopers would like to meet you as you take your trip, and you’d get more human contact that way. If you’re planning to stop in Denver, let me know! 
Also, maybe see if there’s a dopefest in TX shortly after you move there, or at least a small group of people to meet. It’s always good to meet new people that have been established in your new place for a while.
Also, since you are traveling alone, I’d have someone who is familiar with your route in AK and in TX if possible, and call and check in every once in a while, so people know you’re safe. That way, if you don’t call within a reasonable time, your anchor person will know something’s wrong. A lot can happen on the road.
Good luck, and have a great trip!