OP, I'm really sorry that you're having to do this. Not because it's a private = better/state = worse issue per se, but because nobody wants to uproot their children from a school where they are happy and have friends, especially not in Years 8 and 9, which are pretty awful anyway.
As PP have suggested, is there any way you could keep them there until the end of the academic year? IME there is always a bit of movement over the summer; I have had DC in independent schools for nearly 20 years, so I have seen a fair bit of this. I know several families who moved their DC to state schools at this sort of age, and they were mostly fine. For some, it was a positive choice to move their children at 11/13; for others, it wasn't something they would have chosen.
If you really want to stick with the independent sector, you can remortgage, or take out a loan, or sell your house, or pay at least some of the fees on credit cards. I know people who have done all of these (I did two of these myself so as not to upset the apple-cart when the DC were already upset enough by divorce).
If your DC do have to move schools, PP are right that you need to be positive about it all. But I think it's also okay to tell them that you would have liked to have kept them at the schools they are at, but it's just not possible and that you will do your best to find alternative schools which will suit them.
Whatever you do, you are absolutely not a failure. Sometimes circumstances mean you aren't able to do what you wanted to do. That's no reflection on you, though.