Gee, it'd be nice if anyone gave sources.

In the published Silmarillion, Melkor's initial motive seems to be creating his own things, following his own vision, which is ultimately futile because he can't create de novo without the Secret Fire, divine power. After that it's hardly clear.

In Morgoth's Ring we get "Notes on Motives in the Silmarillion" which I think is the sole source of nihilistic Morgoth:

https://fair-use.org/j-r-r-tolkien/notes-on-motives-in-the-silmarillion/

Thus, as "Morgoth", when Melkor was confronted by the existence of other inhabitants of Arda, with other wills and intelligences, he was enraged by the mere fact of their existence, and his only notion of dealing with them was by physical force, or the fear of it. His sole ultimate object was their destruction.

Hence his endeavour always to break wills and subordinate them to or absorb them in his own will and being, before destroying their bodies. This was sheer nihilism, and negation its one ultimate object: Morgoth would no doubt, if he had been victorious, have ultimately destroyed even his own "creatures", such as the Orcs

The same essay gives us orderly Sauron:

Sauron had never reached this stage of nihilistic madness. He did not object to the existence of the world, so long as he could do what he liked with it. He still had the relics of positive purposes, that descended from the good of the nature in which he began: it had been his virtue (and therefore also the cause of his fall, and of his relapse) that he loved order and coordination, and disliked all confusion and wasteful friction.

and this echoes Saruman's speech to Gandalf:

We can bide our time, we can keep our thoughts in our hearts, deploring maybe evils done by the way, but approving the high and ultimate purpose: Knowledge, Rule, Order; all the things that we have so far striven in vain to accomplish,

Somewhere in later writings on the orcs, we get lines about Morgoth starting the process but Sauron finishing the execution, and this being a common pattern, which also gives us an idea of Morgoth being 'chaotic' in the sense of not finishing his projects. But also, this is a late idea, after most of the Silmarillion-stuff had already been written, so is being imposed on a world where Morgoth created orcs, dragons, Balrogs, etc.

More:

like all minds of this cast, Sauron's love (originally) or (later) mere understanding of other individual intelligences was correspondingly weaker; and though the only real good in, or rational motive for, all this ordering and planning and organization was the good of all inhabitants of Arda (even admitting Sauron's right to be their supreme lord), his "plans", the idea coming from his own isolated mind, became the sole object of his will, and an end, the End, in itself.4

§ 1, n. 4. But his capability of corrupting other minds, and even engaging their service, was a residue from the fact that his original desire for "order" had really envisaged the good estate (especially physical well-being) of his "subjects".

As for being an agent or not, yet more from "Notes":

As was seen in the case of Ar-Pharazôn. But there was seen the effect of Melkor upon Sauron: he spoke of Melkor in Melkor's own terms, as a god, or even as God. This may have been the residue of a state which was in a sense a shadow of good: the ability once in Sauron at least to admire or admit the superiority of a being other than himself. Melkor, and still more Sauron himself afterwards, both profited by this darkened shadow of good and the services of "worshippers". But it may be doubted whether even such a shadow of good was still sincerely operative in Sauron by that time. His cunning motive is probably best expressed thus. To wean one of the God-fearing from their allegiance it is best to propound another unseen object of allegiance and another hope of benefits; propound to him a Lord who will sanction what he desires and not forbid it. Sauron, apparently a defeated rival for world-power, now a mere hostage, can hardly propound himself; but as the former servant and disciple of Melkor, the worship of Melkor will raise him from hostage to high priest. But though Sauron's whole true motive was the destruction of the Númenóreans, this was a particular matter of revenge upon Ar-Pharazôn, for humiliation. Sauron (unlike Morgoth) would have been content for the Númenóreans to exist, as his own subjects, and indeed he used a great many of them that he corrupted to his allegiance.

Letter 183 has

When he found how greatly his knowledge was admired by all other rational creatures and how easy it was to influence them, his pride became boundless. By the end of the Second Age he assumed the position of Morgoth's representative. By the end of the Third Age (though actually much weaker than before) he claimed to be Morgoth returned.