All this claptrap about the oversized penises of dead men being preserved after death is ridiculous. I’m a Registered Nurse. All men’s penises shrivel up upon death, just as live men’s penises shrivel up when it’s cold. All men’s penises look to be the same size when they’re dead and if it’s cut off and preserved, no matter the size in life, it would look pretty much the same—a small lump of flesh. You can’t judge the size of a penis if it isn’t erect.
Not men who were executed by hanging:
I don’t know. What did King Saul want with 200 Philistine foreskins?
I mean, why not just chop the whole dick off the corpse? Wouldn’t that be easier?
I have no experience with the penises of dead men but as a gay man who has seen a good number of penises I have to say your last sentence Lulu8360, isn’t exactly correct. I can assure you I have personal experience with a handful of male organs that were impressive in size even when flaccid. Most are “growers not showers” as the saying goes but not always.
There are multiple accounts in The Comedians by Kliph Nesterhoff recounting different people who saw Milton Berle’s infamously large penis. The stories typically recount Berle unzipping and it “flopping” out of his trousers. Flopping is not a term I would imagine using to describe the typical flaccid penis.
Not sure I’d want to wind up on your hospital ward. 
Those two opening lines of your post are candidates for the Straight Dope Top Ten Of All Time. 
I’ve heard that when paramedics arrive at the scene of an accident, this is one thing they look for in male victims - that it may be indicative of a spinal cord injury. Unfortunately, if that’s the case, it’s often the last “natural” one he ever has.
Penises also plump up and enlarge in embalmed bodies.
As told in the book Live From New York:
When Milton Berle hosted Saturday Night Live, writer Alan Zweibel aasked to see his famous penis. Berle unzipped and laid it on Zweibel’s desk, looking "like a life size pepperoni (Zweibel’s words)… Gilda Radner walked in on this scene, said “Excuse me” and walked out.
Then there’s the old joke about Milton Berle and Forrest Tucker.
I just sent that to Atlas Obscura. Could you imagine what Dave Barry could have done with a place like this?!?!?
I visited there last summer. It is a small museum but it grows on you.
The human specimens on display were not impressive.
Make a wallet that becomes a suitcase when you rub it?
Is that the one that involves putting?
From what I’ve read, it involves a bet and Jackie Gleason.
The putting thing is a different gag.