Spoilers alert: This answer will contain spoilers for those who haven't watched season 6. Due to the nature of the question, the whole answer will have to be enclosed in spoiler tags which is something I'd rather not do. Therefore read ahead on your own hazard.
Daenerys was not with Drogon when she was abducted by the Dothraki.
Afterwards, she was taken to Vaes Dothrak where she managed to kill the Khals and unite the Dothraki under her command.
In the episode, Blood of my Blood, she starts moving towards Meereen. At some point in her journey, Daenerys senses presence of her child Drogon and goes away for a while. When she returns, she returns on Drogon's back. The scene is pictured below:
She then arrives in Meereen with Drogon to meet with her besieged councillors.
Afterwards in Battle of Bastards, she defeats the Slaver fleet while riding Drogon and her Dothraki cleaned Meereen from Rebels.
So it appears, Drogon spent sometime healing from his wounds at an undisclosed location and then rejoined with his mother who was already going back to Meereen. He is well and alive, and as of The Winds of Winter, flying back to Westeros with Daenerys and her invasion fleet. Pictured below, Drogon and his siblings flying over Targaryen fleet on its way to Westeros: