So In your scenario the men of the Three houses of Edain had also accepted and worshiped Morgoth?
Morgoth's power wasn't centrecized. It was found in equal propotion in every partical. I am not sure the acceptance of Morgoth as lord would have spurred the marring of their souls.But I think that their acceptance of Morgoth is what spurred the corruption of their fëar and hröar. Didn't Eru say something to that effect as the Voice? That since they accepted Morgoth they would soon die and come back to Him, Eru? As if their deaths is a consquence of their worship...
In which book is the Quote you are refering to?
Sauron is different from Morgoth. Besides Sauron is a Necromencer. He has control over spirits and with his arts Might be able to give them a bodily form.Picture, say, the Mouth of Sauron; that's what was in the future of Men without death to cut them short. And death is thus a very great gift of Eru, a way for them to get away from their enslaver, Morgoth. Eru could not really "unenslave" them without breaking their free will. Men could repent and turn to Eru, but their fëar were already marred, as a result of men's free choices.
And In the case of Mouth of Sauron, he has by his own freewill acepted the virtual slavery of Sauron; but in the case of the whole race of Men in general It cannot apply as so.