What are the pros and cons on using Army movers or moving my household goods myself with a U-Haul?

I'm into letting it be someone else's problem. Think about it. You worked hard for a few years, and get some inherent down time before starting a new job. It's imperative you take advantage of it considering the long game. You'll actually be a better and likely a more promotable asset for it. Those who can't/won't manage time or where their effort is expended, won't do as well as those who do. You don't want to show up at the next base already being beat up and looking it. Oh, you did your own move? Dumb A. Just self carry the stuff you need to hit the ground running. Take the basket day thing to receive HHG, and move on.

Use Transportation/HHG. It's not worth the hassle unless you have very particular means to transport items, 98% of the time, you won't have any issues with the movers... and if you do, you can file a claim, which gets handled pretty expediently, at least in my case, a damaged item which was quickly paid for, so overall I haven't had a bad experience with Army movers. Plus, they do all the hard work, you just gotta make the appointment, wait the 2-4 hour window and then sign the paperwork when they finish packing everything up.

I just did a DITY move, I can tell you now, after having completed one it is not really worth it. The added stress on family, plus driving cross country in a uhaul driving 65 mph is not fun. I suppose it's all on how you go about it, just make sure you don't rush yourself.