I'm guessing by your SWH, that your really just concerned that she's drinking too much and it could be unhealthy (or at the very least, a major pain in the butt because of all the diaper changes) rather than a problem with the sippy cup itself.
My 4 year old drinks a lot, as well. He would go through phases where he drank more than usual. I will tell you that during those phases, I used Huggies Supreme (in the red package). It really helped. We usually used just the regular Huggies, but during those times the regular Huggies would fill up so fast that we wouldn't just be changing diapers we would be changing clothes. The diaper bag had to have extra diapers and clothes. Using the more absorbent diaper really helped. It also really helped at night, when he would really fill a diaper. He went through so many pairs of pajamas!
Since drinking water is just so healthy, you might think about finding a more absorbent diaper to help you through this phase. I'm not suggesting you leave her for hours on end without changing her. But if you find he right diaper, you might be able to stretch out the diaper changes just a little bit.
Hang in there! It won't be that long before she is mature enough to potty train.