If you’re looking at Carleton and St. Olaf, add Macalester to the mix. It has a nominal religious affiliation (Presbyterian), though I think the school feels more like a secular school. However, they do offer merit aid, and the location in the Twin Cities will open up lots of opportunities for internships and applied learning in public health, along with possible shadowing opportunities at local hospitals. You can also cross-register at the University of Minnesota. More information here: Health Professions Advising - Macalester College
And here: Macalester Life as a Pre-Med Student
Just to reiterate my comment above: you are likely to be able to find a faith community of your choice on practically any campus. There are usually interfaith centers that sponsor both interfaith and faith-specific clubs, services, and activities. If a school seems to offer much of what you want and has a vibe that appeals to you but does not have a religious affiliation, do yourself a favor and investigate further. I suspect that the chaplains or interfaith center staffers will be happy to put you in touch with a student to talk about spiritual life on campus (any campus).