All of my crabs are dead when I boil them.
But I eat Dungeness crabs, and I kill them and clean them immediately before cooking them. 
To add more than likely not from Maryland
I grew up crabbing the Chesapeake and Delaware Bays and their estuaries.
Kept wet and shaded, the Blue Claw Crabs will live for hours. You need to move them to ice temporarily or permanently after a few hours. Dead crabs go limp. You’ll know them quickly; you’ll discard them. Healthy ones stay for hours in bushels… and are often transported a hundred miles inland and sold live. It’s common to buy live, road-side Blue Claws 100 miles in-land in summer.
That speaks to their durability. Some won’t make it. You toss them.
For pros, their crabs are kept in pots, and they harvest the pots/crabs, but in bushels, moist and shaded, you have hours to go.
For really long durations, you drop the crabs you caught back into the water, in buckets, lidded with holes drilled in it. Now you can burn 12 hours without concern, then you haul them in and have several more hours out of water before they’re dead.
You can freeze crabs if time is running out and you opt to not cook them.
40+ years of crabbing with trot lines, pots, boats, piers, etc. (Heck, I’m 48. Been crabbing since I was 4… so I’ve been crabbing… too long!)
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There’s been no moderation in this thread. And unless someone decides to report some posts, there won’t be.
Now back to discussing parts of crabs you shouldn’t eat. I vote for all of them, but I suppose that doesn’t really meet GQ requirements.
One question is whether to eat the Blue Crab mustard. This webpage has info, but seems out of date:
I love the stuff, but I only get the chance to eat crabs rarely nowadays.
I read this as a whoosh, FWIW.
There is a willow grows aslant a brook,
That shows his hoar leaves in the glassy stream.
There with fantastic garlands did she come
Of crowflowers, nettles, daisies, and long purples,
That liberal shepherds give a grosser name,
But our cold maids do dead men’s fingers call them.
There on the pendant boughs her coronet weeds
Clamb’ring to hang, an envious sliver broke,
When down her weedy trophies and herself
Fell in the weeping brook.
Always wondered what the shepherds called them.
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Well, here’s a hint: the plant in question is widely regarded as most likely being Orchis mascula, the early-purple orchid, whose scientific name translates as “virile testicles” (probably because of the shape of the root). According to this guy, common names over the centuries have included “dogstones” (where stones = rocks = balls), “dog’s cods” (think cod as in codpiece), “cullions” (Merriam-Webster: “from Latin coleus, scrotum”) and “fool’s ballocks”. I don’t think anyone knows which of these Shakespeare had in mind, but any of them would suit his sense of humor to a T…
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Freshly dead crab may be perfectly fine from a health perspective, but in my (admittedly limited) experience it seems that crabs/lobster/crayfish/etc that were kept on ice postmortem for a couple of hours are never quite as good in flavor or texture as those that met their maker just before dinner. Might be psychological, though…
My experience is with West coast Dungeness crab. Both as a sport crabber and a former employee of a reduction plant that used to process crab shells and waste.
Crabs tend to ammoniate very quickly once they are dead. By ammoniate I mean putting off a level of ammonia that smells like mom mopping the kitchen floor. Hours, not days. This is what kills the other crabs on shows like the Deadliest Catch. A dead crab left in the hold will ammoniate and kill the other live crabs. We aren’t talking about a similar seafood product that is dead but has been held on ice. Fish are fine, crabs destroy themselves internally, very quickly.
I would never cook a “recently dead” crab, only those that are alive when I cook them.