1) Every e-commerce company ever will tell you to offer "Free shipping" even if you have to build it into the price of the bottle. Your target customer doesn't care about the difference between $15 and $20 but, oddly, she does care about paying five whole dollars to ship something she could "just pick up at the store."
2) You're currently selling authentic, which is better than selling nothing at all, but authenticity is not the primary driver of the food purchasing decisions of upper-income Americans. Some options which would complement authenticity: exclusive ("not like the kind you get at the supermarket" / (generally not explicit but heavily implied) "better than what the poor people put on their caprece"), healthy (oh God is that a big one), decadent, social conscience (produced by small family-owned farms rather than big evil agribusiness) etc.
Yes, there's a bit of tension between someone wanting to use class consciousness as conspicuous consumption, but meet your customer where she is at -- and she is at Starbucks.
3) She doesn't care about your story. She cares about her story. Does this make her better than the other moms of the PTA who buy $5 balsamic at the supermarket? If so, lead with that, support with your story.
4) Get some photos of people on your site. Laura of Laura in the Kitchen, for example, as she is a good stand-in for your customer's ideal self. Failing that, if someone who is involved in actually making the balsamic has camera appeal, use them instead.
5) Do some deep thoughts on where people are in the purchasing process when they find you. If they're sure they want this, you don't put the "BUY OUR STUFF" nearly front and center enough. If they're not, your current focus on persuading them that you're the best balsamic makes more sense.