I don't think any of them have to, but it makes more sense for them to move than not
Boston obviously has a major cap crunch than they can solve other ways but getting out on the goalies makes a bunch of sense
Calgary have a cap crunch where moving Vladar would help, and also has a hotshot young goalie to replace him with. They can keep Vladar and send the kid down, but that's not ideal
Anaheim, who knows what's in their head, but I can't see getting Gibson getting more valuable or happier behind their wall of suck
Winnipeg have to trade Hellebuyck between now and the deadline, so if you get a good offer why wait now
San Jose are the only one who seem to be take it or leave it, although even there Kahkonen is a pending UFA who's probably not back, so why not? Maybe you sell him for more at the deadline, maybe not
I don't know any of them become more desperate to sell after the first couple days of UFA, but I don't think they become less motivated. I'm just not sure UFA really affects the goaltending market at all.