Hi L.. My DD was the SAME way!!!
You can try another kind of sippy cup - she might like another brand better.
Also, my daughter really LOVES a straw. Have you tried a sippy with a straw? A lot more fun for the little ones.
Lastly, I kept my LO on a bottle til 15 mos. When she turned a year, she was on 3 bottles a day (I had finally weaned her from Breastfeeding!). At 13 mos I offered her a sippy cup of milk (and other food) first thing in the morning. If she drank it fine, if not - fine, but NO more bottle in the morning.
Then at 14 mos I did the same thing at lunch - no more bottle.
I kept her bottle at night. I actually did not plan on getting rid of that one til 18 mos. but then at 15 mos she actually WANTED her sippy cup at night, and no more bottle.
We took the gradual approach.
Yes, she was drinking less milk, but I added more cheese and yogurt to her diet to make up for it.
I also mix yogurt, calcium fortified juice and fruit, blend it up, and give it to my LO in her sippy and she loves that. She'll drink that over milk anyday.
Good luck!