Depending on your school's resources and your needs, things would definitely be easier with a diagnosis. With my university's "disability" centre, you could get extra time on exams, a note-taker, music/white noise/sound reducing headphones for exams, computers for exams, extra time for projects, CBT counselling, etc. etc.
Maybe it would be easier to talk to your family mainly in terms of it affecting your performance in school? I found college to be way way more stressful than high school (duh), and a lot of my anxiety/"defectiveness" (like, can't concentrate enough on the prof to take proper notes and follow what they're saying at the same time) didn't manifest itself until then. If you have concrete examples of why a diagnosis would be beneficial to you (like, for me, "I have trouble talking to people and asking for help, and would like counselling to make it easier," or, "I find it difficult to concentrate on what people are saying, and would like a note-taker"), especially for SCHOOL, they would probably be more open to helping you out (if they aren't already, if your anxiety isn't only in your head ; )).
No idea about diagnosis price though. I live in Canada and haven't been either. : (