I have exactly the same problem riding my Raleigh Clubman, i was considering fitting more comfortable bars on mine too, the pain puts me off riding it
I reckon you can probably get the stem up an inch, and by twisting the bars round 10 degrees or so raise the brake hoods another inch. That would probably transform your riding position.
Also, if your saddle is still the same position as in the photo, it is sloping too far forwards. You should
try making the top of the saddle as close to horizontal as possible, then if that's not comfortable, make further adjustments from there.
A saddle that slopes forward can cause wrist and/or back pains because you have to use your arms and back to push yourself back into position on the saddle. I had problems with my wrists on one of my bikes and a small adjustment of the angle of the saddle did wonders.