Non-spoilery answer: there's an explanation for this, but you're going to have to wait pretty much until the end of the show to find out what it is. It's not a deus ex machina or anything, though; there are hints dropped along the way, and you may be able to figure it out beforehand.
Here is a fuller answer, which spoils some of the last episode of the show, except sort of not really, since the whole thing is so damned wacky anyway:
The reason that Space Dandy seems so vignettish is that it takes place across a number of different timelines. When Dandy and crew die in the zombie episode, they're dead and stay dead - in that timeline. When we see them alive and well in the following episode, that's a Dandy-and-crew from a different timeline.
You see, the Dandy universe (well, multiverse) is one in which the "many-worlds" interpretation of quantum mechanics (hey, Everett!) is literally true - there are infinitely many parallel universes, and for the most part they can't interact. However, "cosmic strings" (which are probably somehow unrelated to these things) can reach from one universe into another, and the way in which they do is regulated by some mysterious substance called MEGA-pyonium (which, in Japanese, sounds suspiciously like "bigboobsonium").
Anyway, for some reason, Dandy is well-endowed with pyonium ("boobsonium"), which is like MEGA-pyonium, except smaller, I guess? Well, not just "some reason"; apparently, the supreme god of the universe (a.k.a. The Narrator) has had his eye on Dandy for some time, and wants Dandy as his successor for after the universe goes through another Big Bang / Big Crunch cycle, and I guess this is why he has all this pyonium? And, uh, yeah, this doesn't actually matter, since Dandy himself can't really interact with his alternate-universe selves (but then again, there's episode 14...). I think Dandy yanking on the cosmic string in episode 1 may have had something to do with it?
Point of the matter is, there's a lot of universes. And pretty much every episode from 1-25 takes place in a different one. The only time we have any real episode-to-episode continuity is between episodes 25 and 26. In every other case, you can probably safely assume that the Dandy of episode n is a different Dandy from the Dandy of episode n+1.
I should note that I haven't touched the Space Dandy manga; it could be that there are some better explanations in there. (But, given that this is Space Dandy we're talking about, I kinda doubt it.)