Funny you mention Washington University in St. Louis. I have two Southern-California born & raised relatives attending, one as an undergrad, the other in med school. They went to St. Louis for the school, not the climate. And they both feel they very much made the right decision.
My own daughter grew up in sunny and dry Denver - and went to college in muggy and overcast Boston. After her first year, she didn’t even notice the weather. One adapts… especially if one loves the university.
PS… you don’t have to stay in St. Louis for the summer.