tolkiens legendarium - Was Morgoth still a dominant power in Middle earth during the War of the Ring?

Morgoth poured a large part of his hate and malice into Arda. That same tactic was copied by Sauron (good student) when he poured a large part of HIS power into the One Ring. We know the Ring was exceedingly dangerous and always tried to corrupt anyone in possession of it and only truly 'answered' to him. So the question proper is; Was Middle Earth actually under Morgoth's sway and not Sauron's? Even though Sauron himself thought that HE was now the evil overlord, was he just, and still, a puppet of Morgoth's power/influence?

Answer

In a manner of speaking

Morgoth's power was irretrievably sown throughout the fabric of the world; this is basically what we mean when we talk about "Arda Marred"¹. Although Morgoth himself had no direct control over this power (in the same way Sauron can't directly control the Ring when it's out of his possession), it did create a natural inclination towards Evil in anything incarnate².

Tolkien discusses this idea a bit in the (highly informative) essay "Notes on Motives":

[O]utside the Blessed Realm, all 'matter' was likely to have a 'Melkor ingredient', and those who had bodies, nourished by the hröa of Arda, had as it were a tendency, small or great, towards Melkor: they were none of them wholly free of him in their incarnate form, and their bodies had an effect upon their spirits.

^{History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII: "Notes on motives in the Silmarillion" (ii)}

Morgoth was thus actually made captive in physical form, and in that form taken as a mere criminal to Aman and delivered to Námo Mandos as judge - and executioner. He was judged, and eventually taken out of the Blessed Realm and executed: that is killed like one of the Incarnates. It was then made plain (though it must have been understood beforehand by Manwë and Námo) that, though he had 'disseminated' his power (his evil and possessive and rebellious will) far and wide into the matter of Arda, he had lost direct control of this, and all that 'he', as a surviving remnant of integral being, retained as 'himself' and under control was the terribly shrunken and reduced spirit that inhabited his self- imposed (but now beloved) body.

^{History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII: "Notes on motives in the Silmarillion" (iii)}

However

I think it's important to note that Sauron didn't care. He wasn't particularly interested in creating general evil (or full-scale omnicide); he just wanted all of the rational minds of the world under his control:

Sauron, however, inherited the 'corruption' of Arda, and only spent his (much more limited) power on the Rings; for it was the creatures of earth, in their minds and wills, that he desired to dominate. In this way Sauron was also wiser than Melkor-Morgoth.

^{History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII: "Notes on motives in the Silmarillion" (i)}

So although it's true that Morgoth's Evil was still a potent force in the world, it's oversimplifying to suggest that this undermined Sauron's intentions. Sauron and Morgoth were ultimately trying to do different, not entirely incompatible things; Morgoth wanted control over everything, but Sauron was content with merely controlling everyone.

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¹ The phrase "Arda Marred" is used in a few different ways in the Legendarium. Sometimes it refers to the result of the Music, with Melkor's discordant influence (as compared to a theoretical Music that didn't have such influence); sometimes it's used as I use it here, to refer to the world with Melkor's disruption of creation plus the dispersal of his Power; still other times it refers to all of that, plus the physical damage that resulted from many long and destructive wars.

Broadly speaking, you can think of "Arda Marred" as "the world as it is" and "Arda Unmarred" as "the world as we'd like it to be." It's a good enough approximation.

² Which means Sauron himself was not affected by this power; being a non-incarnate thing, his fall was entirely of his own design³

³ Well, okay; entirely Ilúvatar's design