Definitely buy used. You will want a new bike right away in either case.
But go to shops and test ride.
Are you certain you want a road bike? Get a used hybrid with risers for like $250 (like a used sirrus) and go riding. If dead set on a road bike, still spend around that.
That calculator is a bit funky. I am shorter, but have a similar body type. The calculator recommends bike sizes that don't exist. Yours are a bit more extreme, though.
The best I can recommend would be a 58, shop cheap and smart, ride for a bit, then plan the next bike if you like cycling.