I know I should know this, but I don't.

Let's say you have a ship with one hull remaining, and your opponent blasts you at range 1 for 4 hits and you subsequently whiff the defense roll. Obviously you're dead because you suffered more damage than you had hull remaining. My question is this:

Do I draw (or flip, in the case of crits) 4 damage cards, or just the 1 to finish off the last hull point? I've seen this played both ways, but I don't know what the official way is supposed to be.

Thanks in advance.

You draw all cards, and then discard them (Faceup!) once the ship's destroyed.
​If you're clever, that'l help you know what crits are left in your damage deck.

From page 9 of the Rules Ref:

If a ship would be dealt more Damage cards than are needed to destroy it, those cards are still dealt. A ship that is kept in the play area by the Simultaneous Attack Rule is affected by all of its faceup Damage cards regardless of whether they are excess.

and further up...

When a ship is destroyed, its Damage cards are discarded faceup to the discard pile.

I've seen this played both ways, but I don't know what the official way is supposed to be.

DraconPyrothayan and Parravon are right, as they so often are. But as you've noticed, it's a rule that's not followed terribly often--probably because it's not terribly consequential. (In fact, there's a good argument that players who don't deal out and then discard their damage cards are placing themselves at a small disadvantage: they have less knowledge of what cards remain in the damage deck than a player who follows the rule would.)

That doesn't mean you shouldn't follow the rule, of course. Especially if you're at a "formal" or "premier" event, it's important to stay as close to the rules as you can. But I'm a pretty rules-oriented player, and if we were playing in a casual setting, even I probably wouldn't call you on it if you don't deal out all your cards.

The main reason to do it is to prove you've not fixed your deck by removing the worst crits, if too many of a card turn up you know something is fishy.

No need against friends or most people honestly but sadly there are some sad Muppets that have to cheat so in a big event yeah just revel the cards.

The rules would call you for drawing all of the cards. If some of them are face-up and there is a simultaneous fire situation this could actually be important. After the ship dies all of those cards are discarded face up.

From a practical sense there is very little reason to deal any of those damage cards outside of that simultaneous firing situation where the ship remains on the board as a zombie. This is why you will often see it ignored.

Now that "little reason" to mill all of the damage cards you can out of a damage deck is so you can see what's come out of the deck which would give you some idea what may be left. I question how useful that information is because it's hard to predict when you'll get a face-up card and knowing the odds still doesn't promise what the next face-up card will be.

As for checking to make sure the DD isn't "stacked" you should be given the opportunity to do that before the game even begins. While I don't suggest taking too much time doing so I would still say you should take a quick glance through your opponent's DD before shuffling it which you are allowed to do.