No explicit command had passed down
Although it is never explicitly said, it is very unlikely that Sauron "assumed" control of the Dragons after Melkor's exile, however whether he would've been able to rouse them to do his bidding is debatable. Firstly, there were not many dragons left in the time of the War of the Ring. The greatest of which, Smaug, had been disposed of decades earlier. Secondly, of Smaug it is said that Gandalf had thought that Sauron may use "the Dragon" to terrible effect.
Among many cares he was troubled in mind by the perilous state of the North; because he knew then already that Sauron was plotting war, and intended, as soon as he felt strong enough, to attack Rivendell. But to resist any attempt from the East to regain the lands of Angmar and the northern passes in the mountains there were now only the Dwarves of the Iron Hills. And beyond them lay the desolation of the Dragon. The Dragon Sauron might use with terrible effect. How then could the end of Smaug be achieved?
Appendix A, Durin's Folk
The implication being that Sauron could've reached out to Smaug but Smaug was not under his control.
Melkor himself had trouble controlling the dragons and Glaurung, the "Father of Dragons" was said to have revealed himself too early.
Like all dragons, Glaurung was a sentient being, who spoke and took decisions of his own, such as revealing himself too early.
Dragons clearly show, especially in the Third Age, an advanced level of wit and intelligence and a level of sarcasm unparalleled in other creatures. They are shown to posses sentience and as described of Glaurung, made their own decisions.
What use dragons may have been to Sauron is unknown. With the death of the last "Great" dragon, there was not much "fire" left in the others to use to great effect.
It has been said that dragon-fire could melt and consume the rings of power but there is not now any dragon left on earth in which the old fire is hot enough
Fellowship of the Ring, The Shadow of the Past
Tolkien himself states, in letter 144, that the above passage implies that Dragons remained although not in the stature they had seen before
But that implies, I think, that there are still dragons, if not of full primeval stature...
If Sauron had control over what was left of the Dragons, would he not have used them in the War of the Ring in his fight against the Men of Lake-town or against Lorien? While the logistical feat of bringing them to the Pelennor may not have been worth it, using them to attack the more Northern Realms would've been helpful, yet no mention is ever made of that. Nor even a thought in the early development of the writings.