Let's play double jeopardy

The show: JAG
The episode: "The People v. Gunny"
What happened: Gunny Galindez (Randy Vasquez) and his old war buddy (identified for most of the episode only as "Manny," but also wearing Master Sergeant's stripes when in uniform, played by Jesse Corti) leave a noodle house next to a gay bar. A gay man, Edward Proxy (Jamison Jones), innocently bumps into Manny, who immediately assumes the gay man was making a pass at him and retaliates with force. Galindez restrains Manny with the intention of defusing the conflict, but instead Edward takes the opportunity to hit Manny some more. Galindez pushes Manny back and punches Edward back, breaking his nose and blackening both his eyes. Another man comes out of the gay bar to Edward's defense, and it turns out to be Petty Officer Tiner (Chuck Carrington). Tiner is about to punch Galindez but stops when they recognize each other.
After the opening credits, the Baltimore police comes to JAG Headquarters with the intention of arresting Galindez for the gay-bashing incident the previous night. Admiral Chegwidden (John M. Jackson) will not allow that to happen, and asserts that the military judicial system will fairly ascertain Galindez's guilt or innocence. The civilians are not surprised when the judge, Captain Sebring (Corbin Bernsen) finds Galindez innocent. Soon after Galindez leaves the military courtroom, he's arrested by the Baltimore Police Department and taken off-base to be tried in a civilian courtroom for the same gay-bashing incident.
Does it make sense legally? Anyone viewing this episode attentively will be wondering "Can they really do that? Isn't it double jeopardy?" The writers (Donald Bellisario and Larry Moskowitz) anticipated this, so they have Manny visit Galindez at the civilian courtroom and ask about double jeopardy. Galindez replies that he was told that the federal government and the state government are technically two separate sovereignties and that "it's all in the Constitution." Manny is unconvinced by this, and so am I. Neither Manny nor Galindez are lawyers, and neither am I. So I could be missing something.
One can learn something about the law from TV and films, but sometimes one will run into false information. For example, in Bruce Beresford's film Double Jeopardy, a woman is tried and convicted of killing a man who is still alive. After serving some time, she is released on parole. Many people tell her that she can go ahead and kill the man she supposedly killed earlier in broad daylight and she wouldn't go to jail again. Real life lawyer Alan Dershowitz points out that in that scenario the facts of the case would be different: killing someone today is a very different crime from killing someone years ago. Furthermore, the parole board could just revoke her parole so that she serves the rest out the rest of the sentence she got in the first place (when she hadn't yet killed the man). Of course we can't use the film Double Jeopardy to nitpick JAG, but we can certainly use real life American laws.
Let's review what the Constitution actually says. The Fifth Amendment reads:

No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.

The words "nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb" seem pretty clear to me. I'm not quite so sure on whether or not the exception for "cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger" applies only to the need for a grand jury indictment, or if that exception applies to the rest of the clauses in the Amendment. Because if that's the case, then the writers are wrong to have Galindez arrested by the Baltimore police and Chegwidden not once complain about double jeopardy in open court.
The Tenth Amendment reads in whole: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people." Does that mean that an individual State has the power to try someone for something a federal court has found that person innocent? Or if Galindez is found innocent in the Maryland court, could he be taken to a Virginia court, or a Washington, D. C. court, or any other state court to be tried again and again until one of those courts finds him guilty? Or is this Amendment irrelevant to this argument?
To compare an example from real life: If O. J. Simpson was found innocent by a California jury of murdering Nicole Brown and Ronald Goldman, could they have taken him to a federal court in the hopes that the federal jury would come up with a verdict more to their liking? His lawyers would immediately scream double jeopardy. To put O. J. Simpson in jail, he'd have to commit a different crime. Oh, wait a minute, he has (the Las Vegas robbery) and now he's going away to prison for at least a few years.
Anyway, trying Galindez a second time for the same offense is double jeopardy, even if it's in a different court. For what it's worth, that's my non-binding ruling on this fictional case. If you're a lawyer reading this, feel free to let me know if there's some legal subtlety I have overlooked here.