Sauron continued to exist
In a 1968 interview, when discussing if he had met any Sauron characters in real life, Tolkien mentions that he "wouldn’t have liked to meet Sauron when his Ring had been destroyed", thus showing that he felt Sauron still existed.
Of course, there’s another thing, yes, yes, that is a power, yes, yes. People who’ve met some of the great characters [get] a sense of something which is nothing to do with them or their intellectual power. Yes. I wouldn’t have liked to meet Sauron when his Ring had been destroyed, very little exist[ed?]. [But] one does meet people like that. Fortunately the people I’ve met with this extraordinary will-power, which is quite unrelated to [any?] ideas, have nothing. Fortunately when I’ve met them they were fortunately not people with any particular desires, ambition or even sufficient intellect [?].
Lee, Stuart D. ""Tolkien in Oxford" (BBC, 1968): A Reconstruction." Tolkien Studies, vol. 15, 2018
This confirms Gandalf's speculation:
If he regains it, your valour is vain, and his victory will be swift and complete: so complete that none can foresee the end of it while this world lasts. If it is destroyed, then he will fall; and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed.
The Lord of the Rings - Book V Chapter 9 - "The Last Debate"