The Ainur and their physical forms

[obloquy]

Well, since he put a large portion of his power into the one ring, that power stayed intact when his hröa was destroyed. This, I believe, is the only reason he was able to come back the last time. If the ring allowed him to come back, then it could have allowed him to shape shift as well, but personally, I don't think so.

You're mostly on target here. That's exactly why destroying the Ring was the only way to defeat Sauron. The Ring was why Sauron could survive the death of his body once he had become wholly incarnate. Dying as an Incarnate, such as Morgoth did in one version of the mythology (which happens to be the latest version, if I'm not mistaken, as it is included in Myths Transformed, MR), renders the eala (an originally disembodied spirit) utterly impotent and essentially "dead" in the same sense as innately incarnate beings (Men, Elves, Dwarves) died. So this Ring-as-anchor concept only affects Sauron's spiritual persistence, and does not mitigate his status as incarnate; this status being what allows or disallows the act of shape-changing. Therefore on your second point, I would disagree.

Also, it's notable that Sauron was not completely exempt from the negative effects of dying as an incarnate, even with the Ring's support. He still took quite a while to fashion new forms for himself, and he could not hide the evil tendencies of his spirit in his outward appearance.