MB here. The advice about saying it isn't the best fit for you is good. I would look at the total work history you have and understand if you said that the nature of this position has changed and you simply think it's time to move on. More than 2 years tenure is pretty stable so I wouldn't worry about that, and having the nature of the position change is a totally fair (not nasty) way to frame why you're leaving. The needs of kids and families change as the kids age also so that's another way to frame it. What they hired you for in the beginning is no longer what they need, and therefore no longer the best fit for any of you probably.
Re needing/wanting a reference - if you told me you weren't comfortable having your current employer contacted, but you could provide several references from other families (including the one you're currently doing PT work for) that would be fine for me. You can say that you're looking for another position but you're waiting to find the right one so don't feel comfortable having your current employers know that you're considering leaving until you have definite plans. It could take you a while to find the right fit. I would completely respect that (and frankly, it would make you even more attractive as a candidate given the maturity, calm, and professionalism it conveys).
Good luck!