Hybrid or Road Bike?
Firstly, 11 or 12 mph isn't that slow, especially if you're still working on improving your fitness. Try measuring speed in kph instead of mph, as it feels better. You can laugh, but we all do it!

The main thing you should look for in a new bike, IMO is that it fits you. This will improve your comfort. If you're comfortable you'll be able to go faster. Drop handlebars will help with aerodynamics, but probably still less than wearing a less baggy jacket. The weight of the bike is likely to be negligible compared to the weight you've already lost, so don't blow the bank for the sake of a few pounds/grams.

If a road bike fits you properly and doesn't have the handlebars too low compared to the saddle, you ought to be able to get comfortable. That said you can still go pretty fast on a hybrid. Some of my personal records are set on my hybrid rather than my road bike, just because I happened to feel fitter the day I took the hybrid out.

So get a bike that fits and is comfortable. Obviously that means trying them out and making notes. If you are really serious you can go for an independent bike sizing/fitting or follow an online measurement guide rather than relying on the guy in the shop who will want to sell the bike in front of you.

If you're comfortable on your bike you'll enjoy riding and ride more often. This will make you faster. Have fun!