Was Saruman already under the thrall of Sauron?

He does not appear in novel "The Hobbit":

In Jackson's adaptation of the prequel, The Hobbit, Lee reprises his role as Saruman the White, even though Saruman does not appear in the novel. He is shown meeting with Gandalf, Galadriel and Elrond in Rivendell and speaking with them about the mysterious Necromancer encountered by Radagast the Brown, where he tells Gandalf that the Necromancer is just a mortal man and that Radagast is a foolish fellow because of his consumption of mushrooms.

Source: Wikipedia

Even though he is not in the novel, we could try and draw some conclusions from the novels. However, he has a different role there:

Saruman studied in depth the arts of Sauron in order to better oppose him. However, Saruman soon became enamored of Sauron's devices, especially the One Ring. He turned traitor to his mission, and sought power for himself by attempting to find the One Ring. He initially advocated an alliance with Sauron, but he soon betrayed Sauron as well, as his ultimate goal was to supplant Sauron and rule Middle-earth.

Source: LOTR Wikia

So, he never "falls under the thrall of Sauron" in the novels, but rather becomes the victim of his own greed for power.

My final answer to your question on his role in the movies is that we can only try to guess, and my guess would be "yes", not only for his somewhat obvious attempt to undermine Gandalf's warnings about the problem at hand, but also because it hardly makes sense for him to resist the power of Sauron for thousands of years, only to fall under it in the 50 or so years between the events of "The Hobbit" and "The Fellowship of the Ring".

Saruman was not fully faithful to the White Council

Robert T. Tally Jr. analyses the movie's portrayal of Saruman in relation to his more nuanced portrayal in the original Tolkien books.

In his brief appearance in The Hobbit trilogy, he comes off as a querulous, dismissive boss, who is already shown to be subverting the council he heads. For this reason, the “Leave Sauron to me!” — uttered in response to Elrond’s insistence that they pursue the temporarily banished Sauron into Mordor — appears as a dilatory tactic that would benefit Sauron, thus revealing to the viewers (if not yet to Gandalf, Galadriel, and Elrond) that Saruman is already a servant of the Dark Lord.

In The Silmarillion, Tolkien furnishes Saruman's motives at this stage.

Then the White Council was summoned; and Mithrandir urged them to swift deeds, but Curunír [Saruman] spoke against him, and counselled them to wait yet and to watch. ‘For I believe not,’ said he, ‘that the One will ever be found again in Middle-earth. Into Anduin it fell, and long ago, I deem, it was rolled to the Sea. There it shall lie until the end, when all this world is broken and the deeps are removed.' Therefore naught was done at that time, though Elrond's heart misgave him ... Thus the Wise were troubled, but none as yet perceived that Curunír had turned to dark thoughts and was already a traitor in heart: for he desired that he and no other should find the Great Ring, so that he might wield it himself and order all the world to his will. Too long he had studied the ways of Sauron in hope to defeat him, and now he envied him as a rival rather than hated his works. And he deemed that the Ring, which was Sauron's, would seek for its master as he became manifest once more; but if he were driven out again, then it would lie hid. Therefore he was willing to play with peril and let Sauron be for a time, hoping by his craft to forestall both his friends and the Enemy, when the Ring should appear.

The Silmarillion; OF THE RINGS OF POWER AND THE THIRD AGE

In summary, Saruman covets the One Ring and distracts the White Council from the pursuit and absolute defeat of Sauron in order to win a 'long game' by finding the One Ring first.

See also an answer furnished here on Sci-Fi Stack Exchange.