Travelling as a single woman, and as one of a pair of women, I've never had any problems at all (apart from my own anxiety disorder). Everyone has always been friendly and helpful, even to two women sharing a bed in small rural towns. And it's not because I'm open and friendly - I'm very shy and uncomfortable around strangers. Can you listen to your specific fears, rather than just bundling them together as "neuroses" and maybe address them singly so they're not overwhelming? What specific things are you worried about, and how can you ameliorate each fear?