If only we could choose our personalities! I'm sure your daughter, if she had a choice, would be more comfortable with less neediness, too.
No matter what approach you use with her, as long as she knows she can count on your love and approval, she'll probably be okay. Eventually. But if you push her to behave more independently than she feels ready for, she may experience a few months or years of stress and anxiety before she matures enough emotionally to handle it.
Hiding behind mommy when meeting people is probably more the norm than the exception. Crying for a whole hour is a bit more extreme.
You can help her with those separations by introducing them more gradually. How about taking her to the church nursery and staying with her for the whole time for a week or two. Play with the toys and model how she might enjoy interacting with them.
Then find reasons to excuse yourself, telling her firmly that you "need to" go get something, that you "will be back soon," and coming back almost instantly. Bringing back something she likes would be helpful, too – a favorite toy, a binkie, a book, a snack. Then start extending your absences. She will gradually trust that you will come back, as you promised. Drop the "soon" once you extend your absences past 5 minutes, though. That is an eternity for a scared child, and she may become uncertain about what "soon" means.
You can also ask her what things she will do to make time away from you more fun. Ask with confidence, particularly about things she chooses like the dance class. Don't be over-enthused, because kids see right through that ploy. But support her ideas (which may surprise you!) with a cheerful, matter-of-fact attitude.
Once your daughter does get some "okay" experience with being away from you, it may become quite a bit easier for her. Until then, each new situation could be just as traumatic for her for another year or three. She's really not choosing this need, J.. I think you know that, but I thought I'd underline it.