Reach to the handlebars is best adjusted by changing out the stem. How long and how steep is the one you have? Is a shorter or steeper stem possible?

Counterintuitively, sometimes a longer stem can relieve hand and shoulder pressure.

Unfortunately, without access to a good fitter, you're down to trial and error, and unlike moving your saddle, you actually have to swap out different stems. But maybe just maybe your LBS would let you try out some different stems and bring back ones that don't work for you?