My two cents.
Go to your council, a lot of them do a scheme where they help you with a deposit for private renting where you save up and pay them back eventually.
We live an area with one of the shortest waiting times in the UK, a girl I know is a single girl living in a hostel with her daugter, she's been there for 10 months and has only just been offered a scatter house until they find her somewhere permanently.
So if you don't want to go into a mother and baby hostel or something similar I suggest you go private. One bad experience doesn't make them all.
Also, babies don't cost a lot of money if you don't want them to. You can get free and ridiculously cheap everywhere. Especially with the likes of Facebook pages.
As for your dog. I know how hard it can be to give up animals, but you may have to choose between it and your family. Accept that you'll go into a B&B and your partner will live elsewhere, and even then you wouldn't get to keep your dog.