If you try searching for the song titles in iTunes you can play short snippets to see if you like particular versions. If you keep going past the numerous Celine Dion results you should find some good versions. Most of the ones I liked are by musical theatre performers (because that's the music I like!) and a lot are Australian, so hopefully they are available in the US as well.
Beauty and the Beast
- Patrice Tipoki (Australia's Fantine)
- Rob Guest with Tina Cross
When You Wish Upon a Star
- Idina Menzel
- Michael Crawford (the only problem with this one is that I feel like it's the Phantom singing it, which gives it a very different feel!)
- Erin Hasan
- Alfie Boe (who just took over as Jean Valjean in Les Mis on Broadway)
- Rod Stewart
Baby Mine from Dumbo could also be a good choice. There is a gorgeous version by Kate Miller-Heidke on an Australian album called We Love Disney.
(As a side note, I just wanted to say a huge thank you to you for posing this question! In looking up the songs I discovered an album by the late great Rob Guest that I never knew existed! He was an incredible Australian/New Zealand theatre performer who suddenly passed away a few years ago, so it is lovely to have a recording of his voice!)