The shelter that we go to is actually a cat rescue org - they operate a very nicely maintained no-kill shelter, in addition to having cats for adoption at a pet store and in foster care with individuals. I've never been to a shelter that is free of animal odors, though - just so many animals in a relatively small space. DH and I used to volunteer at a no-kill shelter in AZ. So many people needed to keep them running!
If you look at the local humane society's web page, you can see all the dogs at the different locations, and narrow down the ones that might interest you. The other facilities aren't actually that far from St Paul. Also, I know that volunteers walk the dogs from the St Paul AHS around Como Park - maybe becoming a volunteer would be a good option for meeting more dogs/deciding if you really want another one.