This can definitely be done, but I don't think it would be worth it. You could get long-pull brakes levers (to work with V-brakes) and bar end shifters, or get a set of brifters and switch your brakes to cantilevers. You wouldn't need to change your rear derailer, but you'd need to confirm front derailer compatibility with whatever shifters you get.

It looks like the Giant Escape 3 has a relatively long top tube, since it's intended to fit with a flat bar. This means that if you were to switch to drops, you'd likely feel very stretched out, even with a shorter stem. I can't imagine it would be a very comfortable ride.

All that said, I've built up a bike where I switch between drops and flat bars. It seems like plenty of people do make the conversion work. You just need to be a deliberate about your component choices. The comment above about pricing everything out is great advice. I'd say rather than sink money into a conversion, maybe look into getting another bike.

EDIT: Bar ends, like in the post above, are also a great idea to get some more hand positions with your current setup. Nice bike Nao_lin!