But the Second and Third Doctors have very different personalities (and heights!) How would that be explained?
Maybe the Time Lords have Carlos Wu' magical autodoc?
Okay, that's not going to make any sense to anyone who's not up on Larry Niven's Known Space. There's this thing called the autodoc, and it's a coffin-like thing that can heal a person of anything
, including beheading. Yes, seriously. Bey Schaeffer got his head chopped off, the head was put in the magical autodoc, and it grew him a new body that was shorter and not an albino.
As for the change in personality, I think that's overblown. It's a facet of the JNT era, but with the first four Doctors I don't really see anything that one would do or behave that the other three would not.