From bottle to sippy cup...HELP!

My daughter turned 1 about a week and a half ago and doc told us we can put her on whole milk, no more formula, and to transition away from the bottle completely.  I'm totally down with that but my daughter will not drink her milk in the sippy cup!  She has only had juice or water in it and today was our first attempt at cows milk in the sippy.  She wants almost nothing to do with it, like the milk comes out way too fast and chokes her (same with when it's water but she was only taking sips here and there so it was never a concern before).  I haven't given her a bottle all day but I'm concerned she will wake up hungry tonight since she wouldn't drink the milk she would normally have before bed (used to be formula). What else can I do?  Will she just cave soon and accept that the sippy is all she gets now?  She is attached to how the nipple feels on the bottle so I know that's the issue.  Anyone else go through this?