I am sorry this is happening to you. I can't hope to really understand but from the outside, can I suggest that you break things into smaller pieces and work on one thing at a time, or at least start progress and move to the next thing while you wait for a reply. Anything that will require someone else to do something after an an email or phone call is the first thing to get out of the way - you can start progressing and then move to your next task.
First of all, have a look here for how you can get some help with your dogs.
If you or someone you know is struggling to afford pet care basics, help is out there.
www.humanesociety.org
I don't know what's going on with your room mate, but I struggle with the idea that you've both been reduced to zero money in your accounts, they ought to be able to kick in for a few days of dog food. If not, one or the other of you must have a friend who would, even if they held you in very low regard, be prepared to buy a week's worth of dog food.
Next:
Make a fresh enquiry about your disability payments. Explain that your situation has got dramatically worse and that you're facing homelessness. Ask how long it will be until payments begin and ask if there's an emergency fund. Try to get hold of a contact you can go to direct repeatedly.
Work out a plan that ends with you being self sustaining and work out the time it will take, so that if someone is able to step in and help you have a sequence of events and a timescale to work to. Saying "I need to get through the next 2 weeks" or "I've got some work starting tomorrow, and will get paid in 4 weeks time" sounds a hell of a lot better than "I need help until some money appears, but who knows how long that'll be."
Contact any charities who help people with the same condition as you and explain - in a way that doesn't make you sound negligent - that you're in a difficult position.
Speak to your employer, you say that "work has been really bad", I know, but explain you're in trouble and need the work if you can get it.
Have you spoken to your landlord? Do so.
Do whatever you can to avoid homelessness. It will be very difficult to get out of that situation. Speak to any and all family members, even distant ones and ask if it's possible that you could stay with them in the event of the worst happening. Talk to housing charities and explain your situation.
If any of those things are impossible, work out things that are possible.
Treat things like you're in an absolute disaster situation now. Until you've got enough money to sustain food, water and housing for your dogs and yourself for 2 months, you can't do anything but focus on getting into that position.
Talking about needing a good psychoanalyst practically in the same breath as saying that you're thinking of starving your pets to death will exhaust most people's sympathy.