[Long post]
I know that the option I am about to present is not available to just anyone, but should the OP be able to get in, it is viable.
The only organizations of the United States Military (of which I am aware) that will allow you to swear in to active duty, participate in training, and experience the military lifestyle full-time would be the various Service Academies (Army/"West Point", Navy/"Annapolis", Air Force, Coast Guard). As I am about to graduate this summer from one of them, I can offer a little bit of insight into that path into a military career.
I, too, made the difficult decision to enter the academy prior to serving a mission. Barely six weeks out of high school, I found myself immersed in a culture that is often hostile to things spiritual. Abuses of the Word of Wisdom and the presence of pornography are ubiquitous in the living quarters, and that first year I found myself somewhat spiritually adrift. The only things that kept me going were my commitment to attend church, no matter how apathetic I felt, and the availability of a nearby Institute class that buoyed me up between Sacrament meetings. However, I had made a promise to fulfill my commitment to serve a mission, an opportunity that is unique to the academies.
Just a few days before my eligibility ran out, I resigned my spot at the academy and left to serve a full-time mission, along with about ten of my classmates. We were fortunate to be riding on the shoulders of previous classes, who had forfeited their chances at graduating to serve God as missionaries. Their efforts and successful re-applications, coupled with outstanding performances during their post-mission years enabled me and my classmates to return and fulfill our obligations to this nation.
I sometimes wish that I had possessed the maturity present in returned missionaries in my first year at the academy - indeed, after returning, I pulled straight A's and outperformed all of my previous marks physically and militarily. I think that if I had served my mission first, I would also have greatly benefited from the experience and maturity that I otherwise lacked.
On the other hand, my academy experience helped me endure some very difficult trials during my mission, especially when it came to handling the difficulties of apathetic companions and immature leaders. I had patience I would not have acquired otherwise. I had the physical capacity to endure some physical and mental trials that may have sent other missionaries home.
In the end, it's almost a chicken-or-egg debate - would I have been a better missionary if I hadn't gone through my military experience first? Would I have been in the right place at the right time if I had gone earlier? On the other hand, would I have been a better cadet/midshipman if I'd possessed that experience and that spiritual strength?
It's only speculation. And as said before, only the Spirit can help you properly make that decision. Take into account everything the recruiters tell you, especially that those contracts may be overridden far more easily than you suspect. If you go on a mission first, you will definitely receive the blessings. If you enlist, then you are taking a chance. However, you do not know what God has set in motion, and whether or not He intends for you to experience those trials in the military first, unless you ASK Him.
This is a personal decision, made in humility between you and God. Listen to wise counsel, seek blessings, and obey the Spirit. Then, as long as you are faithful, you can know that you are acting in accordance with His will.