Instead of sippies try strawed cups, those are great and my kids loved it. I would put the breast milk in the cups though and continue to offer it to her over and over. If she knows you will cave it will now be a power struggle. You can self wean and wait if she does it but to be honest she could continue to use you as a pacifier and to get to sleep for years! :) She just needs something different to help her sleep by herself. One thing do not nurse her to sleep. Maybe just nurse only in the morning, but at naps and bedtimes try to just take that out of the schedule.
Change bedtime routine now, do something different, have your husband be the last to tuck her in. As long as she can smell the breast milk it will be tough. I did half breast milk half formula when I weaned, then just made it slowly to all formula, she doesn't need formula now at the year mark so do the same with milk and get her used to just drinking milk throughout the day. Have her pick them out and get excited about the new cups.
You need to just decide and prep yourself for a few rough nights. She will cry, she will be upset only because she hasn't learned how to self soothe at bedtime. There is NO way around that, sorry. It may be easy for you or tough, it is all about her. She will readjust her habits with a new routine. It is a habit for her that just needs you to be prepped for helping her break that habit.
Get her a new snuggly toy, change the nap/bedtime routine up, make sure she has a full belly and just prep yourself, you can go in, rub her back if she is upset and snuggle her but I highly recommend your husband do it to lessen the stress for her until your breasts have dried up.
Put it on the calendar, plan it and just do it. It won't be easy, however once your breast milk dries up she will be fine.
Hugs! She won't be traumatized, promise. It is a change, babies and toddlers don't like change but this is the first of many to come and she will be fine.