When you're talking about the flat parts I assume you're not talking about the drops. It sounds like your handlebars are too low.

I'm not a big rider but before I had my handlebars raised my thighs hit my ribcage when I was in the drops. Riding in the drops was definitely uncomfortable. My short arms required a rise of 4cm, not to sit upright but to be in a neutral poisition! My LBS had a very good reputation for fittings cyclists but they missed my short arm issue. I had to go to a Serotta trained fitter to find the problem.

I don't necessarily find riding on the tops unstable, most of the time I'm on my hoods but I do move all around the handlebars. I particularly use the top of the bar for climbing. For downhills I'm in my drops.

How well qualified is your LBS to do fittings? How much have you read about fittings? I just got Andy Pruitt's book and he says a lot about what to look for in bike fit. Be sure you can communicate where you're problem areas are. The more you know about what you want the better fit you'll get.