Spelling it differently doesn't make it less dated.
Jenna derives from
Jennifer so spelling it Genna looks really weird to me, it looks misspelled. I also don't think that
Jenna is awfully dated. A little bit, yes, but not as much as
Jennifer. It is also more dated to the 90s and not to the 70s/80s as is
Jennifer.
I would go with
Jenna Katherine.
Maybe you like
Gemma instead?