By the Third Age, Sauron was having his subjects worship him directly, not Morgoth
Thus, as the Second Age draws on, we have a great Kingdom and evil theocracy (for Sauron is also the god of his slaves) growing up in Middle-earth. In the West – actually the North-West is the only pan clearly envisaged in these tales – lie the precarious refuges of the Elves, while Men in those parts remain more or less uncorrupted if ignorant. The better and nobler son of Men are in fact the kin of those that had departed to Númenor, but remain in a simple 'Homeric' state of patriarchal and tribal life.
c.1951 Letter to Milton Waldman - Letters of JRR Tolkien #131
Sauron desired to be a God-King, and was held to be this by his servants;† if he had been victorious he would have demanded divine honour from all rational creatures and absolute temporal power over the whole world.
[Footnote:] †By a triple treachery: 1. Because of his admiration of Strength he had become a follower of Morgoth and fell with him down into the depths of evil, becoming his chief agent in Middle Earth. 2. when Morgoth was defeated by the Valar finally he forsook his allegiance; but out of fear only; he did not present himself to the Valar or sue for pardon, and remained in Middle Earth. 3. 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.
c.1956 Notes on W. H. Auden's review of The Return of the King Letters - Letters of JRR Tolkien #183