I don't think
Beth is too dated at all! I love it and prefer it over other Elizabeth-nicknames such as
Betty,
Lizzie and
Betsy (though, I do like
Betsy). Bethy would be a sweet pet name to use informally, but not something I'd put on a birth certificate.
Anna Elizabeth called
Annie Beth is adorable. It's cute, but not cutesy, and she could always request to stop being called that if, when she gets older, doesn't like it.