Drax travelled in a TARDIS which he bought secondhand legally, unlike most other renegade Time Lords, who stole theirs. He used it to leave Gallifrey and became a renegade Time Lord involved with freelance engineering. (PROSE: Doctor Who and the Armageddon Factor) It had a working chameleon circuit, unlike the Doctor's TARDIS. At some point in his life, the hyperbolics of Drax's TARDIS broke down on Earth - a common problem, according to Drax, and he was stranded. Drax entered criminal activity while trying to procure replacement parts, and he was imprisoned in Brixton prison for 10 years.
Later after he repaired his TARDIS, Drax travelled to Zeos where he was hired by the Shadow to build the computer Mentalis. He was imprisoned by the Shadow who ordered him to steal the Key to Time, but he had managed to remove his TARDIS's dimensional stabiliser which was later used to miniaturise the Fourth Doctor and himself to defeat the Shadow's plans. (TV: The Armageddon Factor)
From his third incarnation onward, the different incarnations of Drax came together to steal the Blinovitch Limitation Effect limiter, which allowed them all to interact with each other without triggering the Blinovitch Limitation Effect. Since the later incarnations already had the device, they needed to return to their past to ensure they conned the Fourth Doctor, Romana II and K9 into getting it for them in the first place. The later versions of Drax's TARDIS were able to interact with each other just like the different Drax incarnations were; the Third Drax used the appearance of his future female incarnation posing as "Fleur McCormick", who was under the guise of a Galactic Police officer trying to arrest Drax to escape a criminal whom he stole the map to Altrazar off; the version of their TARDIS was disguised as a police car, and hovered some feet in the air near its earlier self.
When the Doctor and Romana believed that they had rounded up all of the Draxes, they handed them over to Fleur McCormick and "half a dozen of her very best men" before they realised Fleur and those men were the rest of Drax's incarnations. They returned to the Rutan's Tendril just in time to watch the police car dematerialise. (AUDIO: The Trouble with Drax)
Behind the scenes Edit
Although Drax's TARDIS is somewhere around in The Armageddon Factor, only components from it are seen on-screen. The interactive Make Your Own Adventure with Doctor Who book Search for the Doctor included Drax as a character, and described his TARDIS as taking the shape of a convertible car, the registration plate of which read: "TARDIS".