ETA after your addition:
Oh my. This is so much more than you originally talked about. This is about your husband making rules for you and not looking at you like a partner in your marriage.
Do you have a family where you feel safe and loved and can go back to? If I were you, I'd be taking a long look at whether or not to be married to a man who doesn't love his child enough to want to take care of her.
This is not the way a marriage is supposed to be. Your husband has no right to order you around. Not being able to afford things is one thing. What is happening here crosses that line, and then some...
Original:
A dog is not going to take the place of a friend. After the novelty wears off, you will be in the same boat. And have more responsibility on your shoulders.
You should look for part-time volunteer work or a job. You might like being a stay-at-home-mom for your child's sake, but you are personally withering on the vine. You are in the same place as many mothers are, especially moving to a new area.
As far as your husband is concerned, he's being wishy-washy and frustrating. He knows that saying no sounds like he's your father. Ugh. So he says yes, but he really does NOT want another dog (he lives in your house too) so he is making excuses about the different dogs. He ends up stringing you along and making you both miserable.
A new dog is not the answer to your problem. You need to get a life outside of your home, and the dog is going to make that even harder. Start now looking for something and finding alternative care for your child. And make it stick, even if you don't really make any money doing it.