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.

Kind of a long drive. We always did a partial DITY move. If you are driving anyway you just need to weigh your vehicle, empty, and then again with your HHG in it. I think you get so much per mile for your POV. As long as you aren't already over your limit for your rank it's a decent way to go.

If you choose to rent a Uhaul type rig, you won't get reimbursed the cost of the truck, but he procedure is the same, and you have at least some of your HHG when you get to your next home.

This is assuming DITY moves are still a thing. Good luck!

I've done both. Back in the day, you could actually make a little bit of money doing it yourself. Now, I don't think the extra time, labor, family stress, and risk is worth it.
It's very stressful on the family to do a move yourself, all the packing, packaging, wrapping, keeping kids out of the way - it's such a hassle.
You assume all the risk for broken items plus your giant vehicle while you're driving. For instance, I had to stop to get gas once. I was driving the largest UHaul with a car dolly on the back. I almost took out a gas pump maneuvering through the gas station.
Unless you have so much weight you have to DITY (like you own a boat) let someone else move for you. They are insured and you don't want to drive across the country after hurting your back moving furniture.

Doing it yourself is generally less expensive, and you have positive control over your stuff (no truck transfers, or combined loads). I’ve done it three time.
If you do the move yourself, check out Penske trucks. They are usually in better condition, and they give military discounts. Whomever you rent from, don’t trust that they’ve done the first echelon maintenance. Before leaving the lot, check all the fluid levels and air pressure in the tires. If you’re renting a diesel for the first time, there is a separate tank for special fluid that’s injected into the exhaust. If it gets low, the truck will slow down and eventually turn off. Make sure they explain it to you.
Last, if you don’t have the manpower and/or time to load/unload your rental, you can hire movers to load and/or unload. You end up doing the driving. I’ve used a local moving company, a third party service offered by Penske, and a semi national company called Bellhops, I believe they are college students. Good luck.