The brake levers will not be compatible with road drops, and the shifters most likely will not be either. You would need new levers and probably shifters, and then the brakes would possibly need what they call "travel agents" to make the cable pull distance work well with road levers. Not exactly a cheap proposition, but not too bad if you can scare up some parts for cheap and feel like a project. The brak cables would also need to be replaced since road levers use a different fitting at the lever end.

What you couold do for cheap is add bar ends to the handlebars to create what are essentially bullhorn handlebars. This will let you stretch out a bit more to get tucked in and have a few more hand positions, but it os not a perfect solution if you really want the drops.