I once tried to do a theatre project for a famous charity. I was going to do all the work (for free) and donate all proceedings to their cause; the focus was going to be the people the charity supports and raising awareness. The charity had to do nothing other than let me put their name on the flyers. I had a meet with the charity but they weren't interested, in fact they were downright rude, accusing me on wanting to ride on their name (which was VERY hurtful and insulting to me because the reasons I wanted to do it was a massive life crisis and me wanting to help those in a similar situation.) A year afterwards, all the celebs started the trend of doing documentary projects for good causes and filmed walks for charities, and nobody complained about them doing it .... Hmmm. What I'm saying, unless you are a well-known person, they may not see any benefit of being associated with you. At worst they may do what that one charity did with me and accuse you of cashing in on them; at best they may just say, OK, send us the cheque with our cut when you're done.
Like Nikki says, being published in their anthologies, or in their newsletters etc is a great opportunity for you as an author to build a career. But it's on their terms and their turf, so doing that it might not help your one book project immediately. I suspect it's more of a long term thing.
Re, selling a specific book together with a charity ... I'm not sure they would physically do much for you. Anything they do would cost them and take away from the cut they get from you. They may be charities, but they need to make money like a business. They want to do as little as poss for you and get as much as poss from you. Their business model is 'other people doing stuff for them for free and other giving donations to them' (basically they take, so they can give to others in need).
I don't really know, but I think doing it with a charity would simply be a case of you doing a lot of marketing yourself, then giving them parts of the proceeds. I'm not knocking that, I think it's lovely, but I don't know where they would promote your book, or what they would do for you. I'm just not sure they would get involved in promoting your book.
But it might be worth a shot. Maybe contact a few and see what they say. If you do you might want to have a plan in place of how you see them help you. Do you just want to put their name on your cover? Or do you want them to list your book on their website? How else do you see the collaboration? I suggest you work out a detailed proposal. Doing that also shows you care for the charity and you understand their mission.
Not sure this helps.