Hello - I'd thought I'd show you mine. I ride in a rather upright style too. If I don't, I get a migraine the following day.
The thing about getting comfortable on a bike, is that there's a number of little adjustments which may not be readily understood.
For example - weight on your hands is not all about handlebar height. Saddle angle is also important. Tilt the saddle nose down and more weight is carried on your hands, no matter how high the handlebars. Tilt the nose slightly tilted up and less weight is transferred to your hands.
Also how far away your wrists are from your body make a difference, too. Cycling with your hands closer to you (longer pull back 'bars/shorter stem) also decreases the weight on your hands, and you don't need your handlebars at such a height.
Equally in regards to your wrists, there could be some other potential solutions. I like a nice swept back handlebars (with a sweep of around 67 degrees) I also like my 'bars slightly pointed down and I like the handlebar grips with the flat bits so I can rest my palms on the grips and they're nicely supported.
Just to echo everyone else here, its worth getting your existing bicycle comfortable first. And it might take some financial investment to do so. You may end up trying three, four or five different saddles, several different handlebar styles and a few handlebar stems. You may even end up changing your seat post.
The other things to remember is the handlebars which you end up liking might actually be good quality but inexpensive, your saddle might be more expensive, but the new seat post may be expensive because its the only one at that length with that layback.