I've been working at lowering my position, and improving flexibility and core strength for about a month now. By now I've figured out my saddle position and feel comfortable with where my handlebars are. They're ~4cm lower and ~1cm farther out than when I started. I found that once I had improved my core strength and flexibility (still working on it though, so close to touching my toes) I could flip and lower my stem without too many problems. I was comfortable as long as I consciously thought about rotating my hips, flattening my back, and relaxing my upper body. The problem was once I started riding hard I stopped doing those things. I found that the key to making it all work was a comfortable seat position and training myself to breath from my belly as mentioned in the thread linked above. Once this happened everything seemed to click.
When I get in the drops I feel like my knees are pushing my chest. At first I thought this meant I'd hit the limit of how low I can go (and that may yet be true) but slowly that feeling has started to go away. I'll keep working at it and see what happens.
All this work has probably made me marginally more aero, my bike look a bit less dorky, and riding way more comfortable.
Thanks for everyone's help.
Keep going! there is obviously a maximal figure to the drop which should not be defined by the frame you ride, either too much or too small. Realistically a 5-10cm range is the most common. Your hip flexors will open up with continued riding and give more power back. In addition to all the stretching and core work, you need to establish a good saddle position, as this is essential for maintaining a rotated pelvis position for drops riding. The right saddle for your body is a key element here, but this can be very time consuming process. Personally I've found good results with cutout saddles and especially Selle SMP saddles. My drop is 13cm on the roadie, but this is in part to maintain flexibility for the TT bike which is my specialty.
Maintaining a proper stretching and core stability routine should be a key part of a riders training to prevent injury and improve comfort. RE: foam rollers, I find that warming up the muscles by 10 mins of gentle riding/turboing, stretching, foam rollering and then stretching again is the best way for me.