I can't help much with bike stability tips except to say that I felt shaky and wobbly at first too. Shaky and wobbly that doesn't get better can be a size or fit issue, though. The bike could be the right size but might not be adjusted correctly for your specific needs. Did you have a bike fit after you bought it?

I'll second the core work and suggest maybe some shoulder strength as well.

I definitely understand being scared of the LBS! To me, most seem very intimidating - very guy-oriented, very Dad's-garage feel, too much stuff that I don't understand, and most of the employees don't seem to realize how to reach out and make them feel accommodating to newbies! Is there someone you could bring with you? Or a beginner group in your area who could "prep" you for the shop (like, suggest a certain person there to approach, or help you with fit questions specific to your bike)?