I'd try changing out the handlebars, as you are already looking at doing. You mention you're comfortable when just your fingertips are on the handlebars, so get a bar with a rise that's about that height (or a new stem, if possible). The brake/shifter cables might still work, try fitting them on before changing the cables. If even one brake cable fits you can at least do a test ride to make sure the rise is correct before changing anything more than the handlebar.
Also make sure you get a handlebar with the correct stem clamp size. Lots of variation in hydrid bicycles, so check if you have a 25.4mm or 31mm stem clamp (or something other than those two).