My daughter had the same problem. She's always been on the lighter side so I was naturally worried about taking her bottles away from her. I didn't think I would ever get to that point where I could give her a sippy cup and that's it. It took a good couple weeks of weaning her off the bottle until now she's 100% off the bottle. I would start by giving her a bottle in the morning and before naps and bedtimes only, so she's still getting plenty, but during the day always have sippy cup filled and available for her to see and play with. She'll get used to seeing it around and she'll eventually be more willing to try to use it.
Also, she's only 13 months, so I wouldn't be too gung ho about trying to completely stop the bottles. This is the age I started weaning so you're on the right track. If she's thirsty she'll drink.
During her best mood times is the best time in my opinion to give her a sippy cup. It'll seem like forever until she'll take it but one day she'll pick it up and that's all she'll want. My daughter went about another 3 weeks or so with just a bottle at night time and just sippy cups during the day. There are all sorts of ways to wean.
Hope this helps.