A good psychiatrist will see you many, many, many sessions (months) before diagnosing a personality disorder. (FYI, I hate the term "personality disorder" because it sounds like a person's character is flawed and that they are inherently "bad.") Personality disorders are diagnosed for people who have patterns of behaviors and beliefs that are so ingrained that treatment is really, really, really difficult. My person feeling is that those with personality disorders actively resist help. In my opinion, they tend to be very rigid in their beliefs, and unlike you, do not see them as irrational. This is why treatment can be a challenge.
I must add that if a treatment program does not work for a person, it does not necessitate a personality disorder. It could be that the treatment was wrong for the person, the psychiatrist/therapist was ineffective, or the treatment program was flawed/bad/damaging.
< personal bias > Or, you could skip the whole psychiatrist thing if you are certain that you don't want meds and begin dialectical behavior therapy (DBT) with a trained professional. DBT was designed for women with borderline personality disorder. A clinician assess you to see if DBT could help you. You usually meet weekly with a small group, and everyone learns and practices new life skills. DBT was found to help other disorders too, like depression, PSTD, anxiety, etc. I have not tried DBT myself, although I am very interested in doing so. It looks more effective to actively attack an issue with practice partners than talking to a therapist ad nauseum. < /end personal bias >
All that said, it is rational to have some fear as a young single woman. You have to be alert. I agree that you may be over-alert, which causes more harm than good. It seems like you have excessive, unnecessary anxiety.
My personal opinion (because you asked) is that you have an anxiety problem but no personality disorder.