To me it always seemed obvious why, however out of character, he was told his "song" was about to end, and it is fair to assume he thought he would die for real, he said "Even if I change it still feels like dying. Everything I am dies" analyse this sentence, he is not dismissing the possibility of regenerating, but it's clear it's a second thought.
I'm fairly sure The 10th Doctor was almost sure he was not going to survive regeneration, if you started with the new series you might not know, but since the classic era, regeneration is bound to fail, it's not common, but it can happen.
Just look at The 11th Doctor just after he regenerated "Legs! I've still got legs!" this could be taken as him being happy that the regeneration was a successful one, in the sense that he had no deformities, but most likely than not, is him being happy that he didn't die during the process.
Now we know as 11th stated "I had vanity issues at the time" and this is true, just listen to 10th after he semi-regenerated previously "I didn’t need to change. I didn’t want to. Why would I? Look at me!" but I doubt this time around, it was him just being reluctant to change his appearance and personality and more about him fearing death.
After the 11th Doctor run, we know he always knew this was his last regeneration, that adds to the possibility of the process going wrong, even if rare, this was his last chance at cheating death, he didn't knew Gallifrey was still out there, he didn't knew there was a chance of getting another cycle, so The Doctor always knew, since 10th redirected his first regeneration towards his hand, that he was getting one last chance, and that would be it.
To me that was always the reason why, the prospect of finally dying, there is nothing wrong with wanting to live, and even if the doctor is semi immortal I don't think he wants to go any time soon, even if sometimes he comes to terms with the prospect of permanent death.
And that was my take on why The 10th Doctor didn't want to go.