Sauron still has a physical form during the events of LOTR. Frodo sees him through the tower window when walking towards Mount Doom, and Gollum remarks he was personally tortured by him, and that his hand has 4 fingers.

I tended to interpret that more like "appearing as a spirit", but you may actually be right. It would explain how he was able to orchestrate and dominate his forces. There is no precedent I'm aware of that any of the Ainur would be able to influence the physical world without a physical presence.

When Isildur slew him, I believe his physical form was destroyed, but as long as a token of his power remained, it makes sense that he would be able to eventually recover enough strength to reincorporate.

Either way, without the Ring, his power was still limited. I'll update my comment, thanks for pointing that out.