Practically the very next passage after Frodo claims the Ring is one where Sauron detects Frodo. He immediately understands that the plan is to destroy the Ring, and with a certain amount of panic, immediately wills the Nazgul to fly like hell to Mount Doom to retrieve it.
And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made; and the magnitude of his own folly was revealed to him in a blinding flash, and all the devices of his enemies were at last laid bare. Then his wrath blazed in consuming flame, but his fear rose like a vast black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung.
From all his policies and webs of fear and treachery, from all his stratagems and wars his mind shook free; and throughout his realm a tremor ran, his slaves quailed, and his armies halted, and his captains suddenly steerless, bereft of will, wavered and despaired. For they were forgotten. The whole mind and purpose of the Power that wielded them was now bent with overwhelming force upon the Mountain. At his summons, wheeling with a rending cry, in a last desperate race there flew, faster than the winds, the Nazgûl, the Ring-wraiths, and with a storm of wings they hurtled southwards to Mount Doom.
Frodo does not appear to have been a lord of anything. He's not commanding Nazgul or Orcs at this point - pretty clearly no one was influencing Orcs, not Frodo nor Sauron, and the Nazgul fly at Sauron's command as always. Frodo's choice to put on the Ring at the last minute rather than destroy it is not portrayed as his heroic will overpowering Sauron's - it's the will of the Ring overpowering Frodo and corrupting him into doing the one thing that will prevent it from being destroyed. Frodo isn't lord of himself at that moment, let alone Dark Lord over Sauron's dominion.
@Werrf's answer appears to have already buttoned up Galadriel's input on the likelihood of Frodo pushing his will on the Ring, but I will add @Royal Canadian Bandit's other contribution from comment here - that we already have one example of a hobbit's will versus Sauron's, that of Pippin's, through the Palantir. Pippin is forced to commune with Sauron, and although he resists somewhat, it is only because Sauron misreads the situation that Pippin escapes without betraying everything, as Gandalf states:
‘All right!’ he said. ‘Say no more! You have taken no harm. There is no lie in your eyes, as I feared. But he did not speak long with you. A fool, but an honest fool, you remain, Peregrin Took. Wiser ones might have done worse in such a pass. But mark this! You have been saved, and all your friends too, mainly by good fortune, as it is called. You cannot count on it a second time. If he had questioned you, then and there, almost certainly you would have told all that you know, to the ruin of us all. But he was too eager. He did not want information only: he wanted you, quickly, so that he could deal with you in the Dark Tower, slowly.