S.-
My daughter had almost this same behavior patter when she was about 3 to 31/2. The only difference that I think makes your situation a little harder is that mine would actually ask for a diaper and once I put it on her she would hide quietly, do her business, and come to me to be changed. We had made a big deal about letting her pick out her first nice "big girl panties" and she was adament about keeping them clean. One day I told her I forgot to get more diapers at the store and she would have to use the potty until tomorrow. With a little hand holding and lots of praise after, she never wore another diaper. Maybe, if you haven't, you could try the panty shopping thing with Lily.
Also, my daughter only used adult potties when we were out for a while. She's very small for her age like me and the big ones scared her for a while. I got a very small training potty that she could move and put wherever she wanted when she wanted to go in private. She was very private about going at first like your daughter and she liked being in control of that. She ended up leaving it behind the bathroom door and from there everything else just came naturally.
I know it's frustrating. It's good your remembering that she is only 3. Hang in there. I hope someone's suggestion helps.
C.