[Azaelia of Willowbottom]

There's no evidence to back this up, but perhaps part of the reason Bilbo started feeling tired and "stretched" could be because he was a genuinely good soul. Perhaps the Ring was wearing him down.

Gollum on the other hand was already a corrupt individual--he had within him the desire to kill for the Ring, and so perhaps he reacted differently to posession of it?

The Ring wore on Bilbo, but in the end his soul was still pure and resilient enough to actually act on the idea of passing it to Frodo (granted, with some encouragement). The Ring twisted Gollum to a greater extent. He became tough and cruel.

Maybe it also has to do with how people came into posession of the Ring. Gollum got it by murder; Bilbo through a less potent combination of accident/fate and trickery.

I dunno. Just a thought, and not a very well-articulated one at that.