Lucky, your Barnet sounds a lot like mine. So stubborn & a fringe, much as I've always wanted one, is impossible.
Please don't chop till you've tried some other options! I've done that so many times & always regretted it.
Jalima read my mind about the bob and Cherrytree about the side fringe at an angle.
Perhaps start with taking off just a bit to get rid of some of the weight ... that can really make a difference in how it feels. Try the side fringe if you like the idea. Then you've got the option of pinning the rest up, plaits, ponytail, etc.
Don't worry about a side fringe not working just because a regular fringe doesn't. You've no idea how my hair resists a fringe, but it loves the side version. The very qualities that make for a rotten fringe can make for a lovely side one - the flippy-factor works in its favour!
If you're not happy then you might want to move on to the immortal bob, which is so incredibly easy and flatters almost everyone.
Big specs usually look good with both the hair-up with side fringe look or bob with side fringe, imo. The urchin cut can look marvelous at any age (like my Mum) with or without specs, but my experience has been that short hair is infuriating. You wake up in the morning & it's sticking out like you've been electrocuted. You're sitting on a chair or sofa, lean your head against the back, or - heaven forbid - scrunch round to the side a bit to look at someone or out the window, and that's it: your hair's moved with you & because it's short it just stays there. If it's longer it moves about better. Doesn't map your every move!
All I care about re. hair is a) that I have some & am v. grateful as I lost masses when I was a child & it was so embarrassing, & b) that it's as easy to take care of as is humanly possible. The simplest things I've found are either: long enough to pin up (with side fringe of course!) or bob (chin length or shorter, just below earlobes) - also with side fringe because otherwise I'm all forehead.
Hope you find something you're happy with!