I think you should tell Josh how it made you feel. Just like you explained it here. It is not about loving the dogs more than him, it doesn't mean that if him and the dogs would be hanging off a cliff and you could only choose one to save, that you would choose the dogs over him.
You need to let him know that you are responsible for them, even if the road gets a bit rocky. Maybe write him an email or a letter and explain it. Don't make accusations and tell him that he would have never asked you this if he truly loved you, etc. If he doesn't care about dogs the same you do, he won't understand, he'll only feel attacked. Maybe he said he understood, or thought he did, about your commitment to the dogs, but never imagined it would go this far. And this is his opportunity to realize and understand it. I can understand completely why you are so upset, but you shouldn't worry that we would bash him, hate him or scream for you to divorce.
And don't give up on the house hunt. Yes, a bunch of places didn't work out, but you also haven't looked for that long. Your husband has a job and you can afford to look at nice places to rent. In the meantime, you have a roof over your head and are not on the streets. I don't mean to minimize your problem, but whenever I feel like I have the worst luck and the world is against me (so to speak), I try to find some perspective to lift my mood.
((((HUGS))))
P.S.: I'll ask my mom if her office can give you some names, they routinely find rentals for employees in the NoVa area.