[Belegorn]

In Morgoth's Ring a passage mentions that "it was the creatures of earth, in their minds and wills, that he desired to dominate." [Myths Transformed] This of course was Sauron's purpose in creating the Rings. It is known that it failed to work on the dwarves and even on the Elves since they took them off as soon as they became aware of Sauron. I think they too could be dominated like Men since they did not dare to wear their Rings so long as Sauron possessed the One.

"in his own Black Land, Sauron made the One Ring, the ruling Ring that contained the powers of all the others, and controlled them, so that its wearer could see the thoughts of all those that used the lesser rings, could govern all that they did, and in the end could utterly enslave them." [Letter of J.R.R. Tolkien, Letter # 131

Tolkien mentions that while in Númenor Sauron did become diminished while corrupting it. He was dominating their wills through the Ring itself. I'm not sure about the Ring, if it's like a battery that can be recharged wielding it's former levels of power.

In letter 200 when discussing the shapes the Maiar put on Tolkien says "It was then destructible like other physical organisms. But that of course did not destroy the spirit, nor dismiss it from the world to which it was bound until the end. After the battle with Gilgalad and Elendil, Sauron took a long while to rebuild, longer than he had done after the Downfall of Númenor (I suppose because each building-up used up some of the inherent energy of the spirit, which might be called the 'will' or the effective link between the indestructible mind and being and the realization of its imagination)." I think it might be true to some extent that when Sauron was overthrown he had not quite put everything back together, regained his hold on things as they once were. So the two kings probably took out a less than 100% Sauron, still a major feat in any case.

Regarding the remaining Dúnedain after the fall of the Kings, none of them had a one among them who could hope to claim kingship as it is said that the line failed in the South. When the Stewards took up their roles "to hold rod and rule in the name of the king, until he shall return"[Appendix A: The Stewards] those who had any legit right had either fled with the other Rebels south or had become much diminished in marrying non-Dúnedain women.

"the descendants of the kings had become few. Their numbers had been greatly diminished in the Kin-strife (about 600 year prior); whereas since that time the kings had become jealous and watchful of those near akin. Often those on whom suspicion fell had fled to Umbar and there joined the rebels; while others had renounced their lineage and taken wives not of Númenórean blood.

So it was that no claimant to the crown could be found who was of pure blood, or whose claim all would allow; and all feared the memory of the Kin-strife, knowing that if any such dissension arose again, then Gondor would perish." [Gondor and the Heirs of Anárion]