In universe....I don't think he does. "We" just don't see it. A LOT of episodes have a line where some characters are TALKING about Odo shapeshifting "Oh he caught the Nassican Smugglers by turning into a three headed Karmdozian razor hawk".
Out of Universe...sure the cost was a factor. Plus a lot of it looked bad...the horrible 90 computer animation.
*Though the BIG problem is: Odo is a Demigod. Just like Data. So the Writers have to forget about him. Odo is immune to physical weapons, so they need to forget that. Odo can't be knocked out with the 'magic TV off button'....but he, of course is..wink wink. Odo can escape and spy easy. And win every fight. And Remember Odo can shape change PERFECTLY into anything except a humanoid.....
The whole "River of Time" and "Set Events" and "Destiny" in Time Travel goes back to..well forever. And it has already been part of Star Trek from TNG on.