Why do most recent Medal of Honor recipients leave the military shortly after being awarded it?

Probably because they get tired of being used in the dog and pony show.

Obviously can't speak from experience, but I suspect that:
1) They've all been through a traumatic significant emotional event
2) Their status as a MoH recipient has moved them away from being with Soldiers and more towards public speaking engagements and poster boy stuff (may not be the comfort zone)

I think the biggest reason would be that MoH recipients from the GWOT campaign were all medically retired, or were already out, Dakota Myers was already out when he received his, Kyle Carpenter lost an eye during his actions that earned his, one soldier lost a hand in a grenade blast, this isn't like Forrest Gump where he receives one and the only wound he gets is a bullet in the ass, these service members get seriously wounded.

Like all generalizations (including this one) this statement is erroneous on its face. But several have already cited reasons that a lot of MoH recipients do not remain in: 1) the award was posthumous; 2) they received serious injuries; 3) they never intended to be career anyway. On the other hand I have know two who both remained in long after the award. One was awarded the MoH for actions in Korea as TSgt (USMC had that rank in those days). When I met him, he was a CWO-4 (prior to CWO-5 creation) and a "bursting bomb" Marine Gunner. That was some 20 years after his award. He was pretty "salty" (deservedly so) and when I asked if he always talked to the CG the way he did on the phone while I was sitting there, he responded, "Hell, Skipper, there ain't no W-5 board this year!"

The award is very rare. The only time I have ever seen a man wearing the award in person was of all places Iraq. I had just cleared my weapon and turned the corner behind a concrete barrier into the chow hall at TQ when a tall civilian crassed before me. Around his neck was a strange baby blue scarf I could not see from far away. Suddenly the bronze gold colors of the star afixed to the ribbon came into focus. The man was my father's age a Medal of Honor recipient from the Vietnam war. Now I had a real problem, there was a MOH clear as day but I was in a war zone, do I solute or not. I decided the jersey barrier wall provided the man enough cover so I came to the position of attention and rendered a salute. The man humbly thanked me and sook my hand. When I entered the chow hall i found several other men all with the same aware around their necks. It was a stange occurrence to see these men all in their sixties over there in Iraq to support us, their generation of veterans is one of great humility and fortitude, thank you gentlemen.

Not all do. In Air Assault School @ Campbell in '75, graduation day, and I came around a corner at a high rate of speed, almost knocking over one of the instructors. First thing I saw was this leeetle baby blue ribbon with white stars. Stopped, rendered a salute, and moved out. It was so unexpected, and shocking that I have never been able to remember his name to my regret.

In a similar vein, in the civilian world, I worked with another recipient at a job I won't mention so he won't be identified. Because of his acclaim and ability, he was given a good position and did his job as expected. A new plant manager took exception to what was being done, and Fired him. He laughed as he walked off the property, returning in a day or two to a new plant manager :)

It gets difficult to continue to work, carrying around such big balls.

Not everyone joined the military to be a careerist. Being awarded the MoH doesn't change that. The same reasons people stay or get out still apply to MoH recipients. BTW, the reason why it seems that MoH recipients all get out after they are awarded it is because a) the medal is so damn rare that there is just not enough of a population to get a good generalization regarding whether they tend to stay or go....and half the time the medal is awarded posthumously which decreases the population even more, b) many are so severely injured they HAVE to retire from the service, c) many recipients had already gotten out by the time their medals were finally approved. Then there are the reasons already mentioned (pressure from the status, trauma, etc).

I think a lot of people leave the military because they end up medically retiring. If you find yourself with an MoH, my guess is theres a good chance you also have a Purple Heart. So they might not have the choice of remaining in. In fact I think Ty Carter petitioned to come back in. The MoH recipients might not finish their twenty because they are able to retire earlier.