Ggtkuk

I’ve recently been doing some builds for Necromancers for a one-shot we’ll be doing. I have been struggling to find anything that mentions how to keep up with all of the corpses you will need for the Animate Dead and Create Undead spells.

There seems to be an indefinite period a corpse can stay a corpse within the 5E RAW.

I also haven’t found anything stating that undead that are killed again cannot be animated again, within RAW.

Once my zombie or skeleton dies and becomes a corpse again, may I reuse the now corpse according to RAW?

Maybe for skeletons, but other undead probably can't be re-reanimated

It's important to note that the spells you mentioned can create undead minions out of a specific type of remains: not just any kind of corpse can be animated by these spells (you can't raise an undead dragon or minotaur with them, for example): they are specifically designed to create undead out of dead humanoids.

The spell Create Undead contains the following text (PHB p. 229, bold added):

Choose up to three corpses of Medium or Small humanoids within range.

The spell Animate Dead has a similar requirement (PHB p. 212, bold added)

Choose a pile of bones or a corpse of a Medium or Small humanoid within range.

It might not be obvious why this matters at first. It might seem that if you raised a zombie from a corpse of a humanoid, and then that zombie was destroyed, the resulting new corpse is still a dead humanoid. But a destroyed zombie or ghoul is not a humanoid corpse: it is the corpse of an undead!

As such, most destroyed undead are not valid targets of the animate dead or create undead spells.

Why might skeletons be an exception?

The text of animate dead is somewhat grammatically unclear about its possible targets. At first glance, it seems that the "pile of bones" or "corpse" must both have belonged to "a Medium or Small humanoid." But if we delete the term about the corpse, the resulting sentence:

Choose a pile of bones... of a Medium or Small humanoid within range.

becomes grammatically awkward. The pile is the thing being said to be "of" the humanoid, which is strange since humanoids don't contain piles of bones. If they had said "a pile of the bones of a Medium or Small humanoid", or "a pile of bones from a Medium or Small humanoid", it would have removed this grammatical awkwardness (by stressing that the bones, not the pile, are the thing once inside a humanoid). But this is not the structure of the sentence we are given.

However, there is one other way to remove this awkwardness, which is to assign the "humanoid" requirement only to the corpse, not the pile. This makes the sentence regarding the pile of bones less awkward, like so:

Choose a pile of bones... within range.

This is a grammatically valid reading of the original sentence, given its ambiguity (and the grammatical strangeness of the other reading). Thus, you could conclude that the bones could come from creatures other than humanoids, such as an undead.

Note that your DM may take issue with this reading. After all, it could permit you to animate a skeleton from a pile of bones from a mouse or a dragon, which is likely outside of the scope of the intended use of this spell. And since the RAW on this issue rely on an ambiguity, your DM will need to sign off on this. But if they do, you could possibly use animate dead on a destroyed skeleton to restore it to a foul semblance of life once again. After all, although the pile of bones are no longer the bones of a humanoid, they are certainly still a "pile of bones."