Yes, hyvästi feels quite old fashioned, and permanent. Maybe older people use it more, or in some Finnish dialects. I think I’ve only used it as a joke, and not very often, or it wouldn’t be funny. 
Sometimes we say näkemiin. When I say bye, I think I usually say “Moikka”, or “Moi moi” or “Nähään” or “Nähdään” or “Nähdään taas.”.