Purely from an in-universe perspective, he really seems to do it less once he learns that his people are the Founders of the Dominion. Perhaps he shapeshifts less to differentiate himself from their fascist rule, or so that few recognize him as a founder. We also see he is uncomfortable with the Vorta and Jem’hadar seeing him as a god.

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.....

For one thing Odo can’t shapeshift for a significant chunk of season 4

My in universe reasoning is that he doesn't shift less, he just does it privately. I don't think he likes to remind people how different he is and he wants to distance himself from the founders who are known for taking different shapes.

My memory isn't perfect but I think after visiting his people he gets a room for the first time, fills it with things to mimic and stops sleeping in a bucket. He has later episodes where he gushes to Lwaxana Troi about taking the different shapes and how fun it is so he's definitely shifting. After he gets his abilities back and starts his relationship with Kira we see him have that crazy sex scene in Chimera, so that's happening off screen a lot.

Also in the episode Chimera he explains to Laas, the other changeling, about humanoids fear of their shape-shifting abilities. It seems like he's keenly aware how different he is and takes steps to cover it up.

Saying that he is still obsessed with the Great Link and yearns to go back so I can see him practicing his abilities in private to keep them up to par for his return. A wish he can't say out loud because of the whole 'evil empire at war with their way of life' thing.

I think narratively it's just a much more enjoyable to not have Odo shift shape onscreen, especially as it gives him the ability to overpower a lot of people. In the early episodes this needed to be shown so we could understand Odo's character, but by the fourth season (both when he was a solid and not) we knew enough about the Changelings that we didn't need to show this as frequently and had to figure out more creative ways to to get around Odo.

I felt as though something was lost when Odo rarely shifted. It became an almost implausible possibility despite being an essential component of his identity.

My bet on the real reason was it's boring and solves too many problems. It also likely cost more than just having the actors do something instead.

There's probably lore reasons you can use, like the one mentioned here about trying to differentiate himself from his ancestors, but I'd bet the real reason is just for better storytelling opportunities.