Then it dawn on me that the problem might be handlebar height since I have very short arms compared to my height. I'm 183cm tall(6'0") and have a wingspan of 175cm (5'9").
Am I out of options here or is there some other way to raise the handlebars? Getting really desperate here. This is also my first "real" bike.
Wow you are way off average when it comes to the difference between your wingspan and your height. A difference of +4 inches is considered 'normal', whereas you are at -3! Consequently, you are what bike fitters would call a very difficult fit (check out the last two videos of the 'bike fit advisor' on YouTube, as a newbie I can't post links yet).
You could try the various stem risers on ebay (something like 60mm 35degrees), but they are better at increasing your stack than shortening your reach. yojim.net has a tool which will tell you exactly how much difference in stack and reach these stems will make (sorry can't post the link!)
I don't want to sound pessimistic, but your best bet might be a custom frame!
