I think it's debatable whether using the ring would still mean you'd fall to Sauron or if you'd replace him as the big bad. I can't remember what the books actually said. Either would be an interesting story. I think Tolkein's works are so archetypal - they literally set the groundwork for the epic fantasy genre - that they're hard to ... riff on? Not so much parody as that's been done, but I can't imagine a Middle Earth where, lets say Saruman successfully keeps the Shire. Maybe that's because I'm not an author.
It would depend on whos' will was the strongest. You could argue that because Sauron(well, The Enemy in general but acting through Sauron) corrupted Saruman that Saruman could never truly challenge Sauron, much in the same way Sauron could never truly challenge Morgoth. However, this is forgetting that Saruman, Sauron & Gandalf are members of the same race or class(not in the RPG sense).
Morgoth, even as the weakest of the Valar by the time he was fully corrupt, could still have smote Sauron in an instant
before Sauron forged the Ring. After that, I believe Tolkien stated that Sauron had done what Morgoth couldn't, namely, he
created something. He used materials and knowledge from others, but it was still original in some way. There was more than pure contempt and hate in its construction, much like how Aule made the Dwarves. They were obviously
of Illuvatar but they still retained a spark of creativity & originality.
The Ring in that regard is how Sauron eclipsed his master. And even this is because Sauron was inherently 'more good' than Morgoth, because according to Tolkien the simple fact that Sauron was
willing to do the work of another(that is, not acting purely of his own self) meant that he had some small sliver of inherent morality & goodness in him. This can also be seen in what little we know of Sauron's actual goals regarding Middle Earth. Morgoth would have "ground the Universe to dust and then hated the dust". But with Sauron, we see a measure of care, however fickle it may be, in his armies and captains. He created a language for them(I believe it was Sauron who crafted the Black Speech, correct me if not) and was willing to dole out power & reward(however little) to those he favoured.
Do note; I am not saying that Sauron with the Ring is more powerful than
Melkor, or even the Morgoth after the end of the first war. But by the time he's chained down outside Arda? That Morgoth would likely be smitten by Ringed Sauron.
TLDR; Sauron
created something rather than just corrupting it. Corruption was involved, but was it not just mockery of someone elses' work. And because of that sliver of good still within him, he would forever be Morgoth's superior in power and influence.