Not quoting Russ's entire missive, but I'm a rider of a certain age, and all my bikes have bars that are no lower then my (properly adjusted) seat. Having the weight more on my seat and not either on my wrists or somehow magically hovering over my wrists while engaging my core has been key. (I oddly feel I should point out that I've got pretty decent core muscles ... but my wrists still need to be no lower then my seat for reasonable weight distribution.)
Maybe make sure that whoever does your bike fitting is older then you are...
(Russ, I also picked a fight over my third chain ring, which I won when I pointed out that 1. I want to be able to ride up Doc Bill's Hill seated, and 2. I'm an older lady, i.e. you can't shame me for not having a big enough member, ifyouknowwhatImean. I can balance on a bike climbing shockingly slowly...)