The question is whether the bike frame is the right size for you- did you get fitted on this bike when you bought it? Assuming you would have been "able" to stand over several different sized frames of this particular bike (because it was a mixte)- how did you know which size frame to get when you got it?

Sandra- though pushing the saddle forward might help in some ways, it can also create other problems. For some people doing that can feel better. For me, pushing my saddle forward actually moved my center of gravity forward, more over the crank, and though it seems counter-intuitive, it made me feel unbalanced and made my weight fall forward even more forward onto my hands.

I am puzzled by your feeling your weight is falling forward, and puzzled by your feeling you need to sit upright more- right now your bars are substancially higher than your saddle already- should be a comfy posture.
The only two reasons I can think of for your having too much weight on your hands at this point are:
Bike frame too long for you.
or
Your bike fits you but- your core muscles are not strong at all or you are riding very little. If you ride regularly (40-50 miles a week maybe) then you will build up your muscles in such a way that they will hold you up nicely with no feeling of falling forward. Even having good quads (top of thigh) will help you get your weight balanced over your legs more and off your hands.

Try to ride with more of your weight on your feet as you pedal. See if that alleviates your saddle discomfort AND your hand discomfort. Remember- riding a bike is not like zooming around on an office chair. See if you can distribute your weight more evenly between your seat, legs, and hands. Work your legs and abdomen. Think of pedaling as WALKING your bike, not like just sitting on a chair and shuffling your feet. This mental image helped me get myself off my *ss while riding my bike. Standing up while pulling up small hills helped me a lot too.
It took me many months of riding before I felt like I wasn't putting all my weight on my butt and my hands.

Maybe I am out in left field but it's just a couple of other thoughts to consider. It helped me.