The control levers and shifters for flat bars don't fit on drop bars; they have different diameters. The clamps for your brake levers/shifters are too small (22.2mm or 7/8") where drop bars are 23.8mm or 15/16". They physically won't fit.

So you have two options:
1) Find a versatile handlebar that takes MTB controls. Trekking/butterfly bars, Albatross/Albastache/North Road styles, Bosco/cruiser styles. I use a cheap Wald cruiser bar on my MTB to townie conversion.

2) Change the controls to road/23.8mm to match the new drop bars. You'd still have to get brake levers with the proper pull (for linear-pull or mechanical disc brakes); Tektro RL520's are cheap and popular.
For shifters, you'd have to get 9 speed bar-ends or thumb shifters. You couldn't use STI brifters; I don't think any 9 speed brifters have the proper brake pull for disc brakes.