I think the answer here is a clear 'no'.
The part below contains spoilers to multiple Harry Potter books.
Tom Riddle himself explained at the graveyard that after being hit by the rebound of his killing curse, he was torn out of his body and used to live in animals and later in Quirrel. He needed unicorn blood to stay alive at that time (as you may remember from Philosopher's Stone).
This points toward saying that two souls cannot live in one object (in these examples, living things) for a long time without hurting or even killing the weaker soul.
In the prophecy it is said that one cannot live without the other dying, this may (I repeat, may as I am merely speculating) mean that Harry or Tom's soul piece in Harry, would die either way even if Harry was to turn away. By giving in to Tom in the end he made sure the right piece, the weaker piece that could not give in, was killed.
This would suggest that Nagini would not be able to grow very old either unless if the horcrux would literally take over. To strengthen this claim, Dumbledore told Harry he suspected the snake of being a horcrux as Tom had exceptional control over the snake, even for being a parselmouth (HP and the halfblood prince). This would be the case if his soul was the one in control.
This same reasoning could apply for Ginny getting weaker while Tom-in-the-diary became stronger and she would ultimately die in order to give him the strength to rise up again (Chamber of Secrets) in the chamber. But as this does not literally contain two souls in one object as Ginny was not split so far as we know (though she may have been partly drained into the diary, we know that Tom-in-the-diary could take over her body for short times at that point).
I am sorry for not using the specific quotes from the books but I hope this answers your question.