The council are very likely to meet with your mother and see if they can change her mind first, and get her to agree to you staying. They did a really hard push on my landlord when this happened to me, and I was 17 and not at all related to him.
Anyway; what it does to your priority status will depend on where you are, I expect - but if you’re looking at going into hostels, I’m going to presume there’s a shortage of housing. That means that they’ll likely not offer anything over legal limits, so you won’t be considered highest priority if you don’t stay in the hostel.
I had to stay at the Salvation Army, and they were very clear that if they found that I was spending time or sleeping elsewhere; they’d consider me as having other options. The doors didn’t lock and it was pretty scary, and I already had very severe anxiety, but I got through it and you will too. I wouldn’t do it again, given the choice, but I’m okay - and it sounds like you’re looking at somewhere a bit better than that, and you won’t be alone.
You are very likely to have to start contacting charities about your dog, though. I think there are some that will look after them temporarily until you can take them back - the Cinnamon Trust used to, I don’t know if they still do.
Have you been told how long you’ll have to wait for more permanent housing? Temporary council housing where I am (Midlands) is for an average of four years at the moment.
Have you looked into the other options? Your council should have a list of local landlords who will accept a bond from the council rather than a deposit, so if you or your partner gets a job (any job, it doesn’t matter what it is right now), that might be an option for now?
If it really is a case of nowhere to go and no prospect of finding somewhere yourself, then it’s worth getting your head around how it’ll go - you won’t have much say in where you live, and you won’t be able to take your dog, but you will be housed and warm and have your partner and it’s so much better than it could be, honestly.
The worst part of this will be the unknown, especially with the anxiety, so make sure that you’ve done all you can to have more control over this yourself; and use lots of coping mechanisms if you are having to take the homeless route - it might feel like none really help by themselves, but it’s really worth a shot.