I'm sorry you don't like your name. I never liked mine for many years either, not that it's so unusual but just because there were no others in my class with my name. I sort of great out of that after being in enough organizations where all the Debbies, the Davids and the Marshas had to be differentiated by using a last initial too.
I think you can choose a nickname any time you want. People who've known you for a while won't use it, at least not right away, but going forward, everyone will. I know a Kathryn who started using "Kat" in high school, and while her parents and sister don't usually use it, everyone else finally caught on.
I think you have to lighten up a little with people who don't get it the first time. Yes, you say your name, but you say it's also unusual. So if the person is an auditory learner, they may get it the first time. If they're visual, seeing it written (or perhaps having it spelled) can be helpful.
I think there are so many interesting names now, as people embrace their ancestry and culture, along with the many people who have immigrated from other countries. I don't think there's nearly the stigma that there used to be about unusual names. I do sympathize with people who have very long names (one friend of my son's couldn't fit either her first or her last name in the letter blocks they allow on standardized tests - just not enough spaces) and I sympathize with those whose names are difficult to pronounce (think of actress Saoirse Ronan, in which Saorise is pronounced "ser-sha" and rhymes with "Inertia"). There are so many celebrities now who have resisted changing their names, and the public is coming along to deal with it. In addition to Ronan, think of David Oyelowo, Chiwetel Ejiofor, Idris Elba, and Uzo Aduba. Some of them have decided they like or are proud of their names.
There are tons of celebrities who have changed their names, for many reasons. Sometimes there is already someone with that name, for example. My husband is a writer but he has to use a different name for his published works because there is already a famous person with the same name.
I also think there's a level of rudeness sometimes among Americans who don't want to bother to learn unusual names, and it's rooted sometimes in laziness and other times in anti-immigrant or racist tendencies). So I think there's pressure you may have felt about being "different" that's rooted in some not-so-nice realities. For many years, a lot of people changed their names to get away from ethnic labels - Jews, Poles and many others changed their names to avoid discrimination. Fortunately much of that is changing. So there are many reasons why people can't (or won't) get the name, some understandable and some unreasonable.
Whether your change your name legally or just go to a different nickname is up to you. If you don't change it legally, then you have to use your legal name on your driver's license and medical forms, although most of the medical forms I see now for doctor/dentist offices ask what you want to be called. Same goes for employment situations - the name on your office door or your email address and so on.
I hope you find a solution that makes you happy.