I'll first state that I haven't done Disney for a long trip with CDs. DS figured out the potty by the time he was 2ish, so our first big trip was entirely diaper-free.
But we have traveled with them!
I will also state that I am vehemently opposed to using disposables. It is a deep philosophical issue with me, and one of the many problems I had with the hospital transfer was that no one thought to bring the dipes, so for 43 hours my boy was in disposables and I hated that.
So I CD even when vacationing.
Although I had all sorts of attempts at complicated washing systems, ultimately I did best with simplicity. One wash on hot. Every so often I would do an extra rinse/spin but it wasn't vital for me. I used a small bit of washing soda and about 10 squirts of Simple Green. That's it. So washing for me was very very easy.
But I didn't feel like buying those things for our first long trip where I would have to wash, so I just used the detergent at my brother's. When we got back I had to do a few extra washes b/c he uses Tide and I'm dreadfully allergic to it, the smell is just awful (I can't hug my MIL and breathe at the same time b/c she stinks of Tide) and I had to get it out, but a wash or two in a different detergent didn't ruin anything.
I had two wet bags (big ones) and when traveling I brought both. One had the clean ones all folded and stacked, and put into a suitcase. The other would get the dirties. Don't put that straight on the carpet, by the way, b/c the bags breathe a bit and can do yucko things to a carpet.

Put something under it, or put it in the bathroom (probably best anyway).
Wash when you have time; when kids are napping one parent can go wash and read a book.
Have fun! You are not alone in not wanting to use disposables.