I agree with the points made by Victim of Circumstance, Jacob C. and Ian Thompson in their answers. I would like to add the following points.

He didn't have the time or resources to try every strategy that we can dream up

Powerful as he was, Sauron doesn't have an infinite amount of time or resources. The search for dragons would have taken resources away from other efforts, such as recruiting the Haradrim and the Easterlings, that he might have thought more likely to bear fruit.

He didn't think he needed additional help to win battles

Also, according to his wisdom, Sauron knew that the forces he had were enough to ensure his victory. While using the palantír, Denethor is convinced of Sauron's certain victory.

‘My lords,’ said Gandalf, ‘listen to the words of the Steward of Gondor before he died: You may triumph on the fields of the Pelennor for a day, but against the Power that has now arisen there is no victory.

 

_{The Lord of the Rings Book Five, Chapter 9: The Last Debate
Page 878 (Single volume 50th Anniversary Edition)}

Sauron comes to realise that the Ring is all that's important

Gandalf explains this to Aragorn, Legolas and Gimli:

The Enemy, of course, has long known that the Ring is abroad, and that it is borne by a hobbit. He knows now the number of our Company that set out from Rivendell, and the kind of each of us. But he does not yet perceive our purpose clearly. He supposes that we were all going to Minas Tirith; for that is what he would himself have done in our place. And according to his wisdom it would have been a heavy stroke against his power. Indeed he is in great fear, not knowing what mighty one may suddenly appear, wielding the Ring, and assailing him with war, seeking to cast him down and take his place. That we should wish to cast him down and have no one in his place is not a thought that occurs to his mind. That we should try to destroy the Ring itself has not yet entered into his darkest dream. In which no doubt you will see our good fortune and our hope.

 

_{The Lord of the Rings Book Three, Chapter 5: The White Rider
Pages 496-7 (Single volume 50th Anniversary Edition)}

I agree with the points made by Victim of Circumstance, Jacob C. and Ian Thompson in their answers. I would like to add the following points.

He didn't have the time or resources to try every strategy that we can dream up

Powerful as he was, Sauron doesn't have an infinite amount of time or resources. The search for dragons would have taken resources away from other efforts, such as recruiting the Haradrim and the Easterlings, that he might have thought more likely to bear fruit.

He didn't think he needed additional help to win battles

Also, according to his wisdom, Sauron knew that the forces he had were enough to ensure his victory. While using the palantír, Denethor is convinced of Sauron's certain victory.

‘My lords,’ said Gandalf, ‘listen to the words of the Steward of Gondor before he died: You may triumph on the fields of the Pelennor for a day, but against the Power that has now arisen there is no victory.

_{The Lord of the Rings Book Five, Chapter 9: The Last Debate
Page 878 (Single volume 50th Anniversary Edition)}

Sauron comes to realise that the Ring is all that's important

Gandalf explains this to Aragorn, Legolas and Gimli:

The Enemy, of course, has long known that the Ring is abroad, and that it is borne by a hobbit. He knows now the number of our Company that set out from Rivendell, and the kind of each of us. But he does not yet perceive our purpose clearly. He supposes that we were all going to Minas Tirith; for that is what he would himself have done in our place. And according to his wisdom it would have been a heavy stroke against his power. Indeed he is in great fear, not knowing what mighty one may suddenly appear, wielding the Ring, and assailing him with war, seeking to cast him down and take his place. That we should wish to cast him down and have no one in his place is not a thought that occurs to his mind. That we should try to destroy the Ring itself has not yet entered into his darkest dream. In which no doubt you will see our good fortune and our hope.

_{The Lord of the Rings Book Three, Chapter 5: The White Rider
Pages 496-7 (Single volume 50th Anniversary Edition)}