All the Doctor's deaths outside of Throughton's have been violent and/or sudden outside of the couple he's held back.
But same result, no time to try on different bodies before settling on one.
If we assume the Ainley Master got a new set of regenerations during his Five Doctors adventure, and he really did die 2 or 3 times after that, he came back with the same look and persona.
Well the End of the War Games, the Timelords were giving the 2nd Doctor an opportunity to choose his next body, making one wonder if their control over Regenration and manipulation of DNA was a technology that was secret. The 2nd Doctor acted quite surprised. Later we see Romana body shopping which makes me wonder what devices she was allowed to bring on board the 4th Doctors ship, as she was already there, and had long enough time to make mods to the console while the good ole doc was parlaying it up with the White Guardian.