i cannot afford this mortgage and all the bills and all the dog stuff on my own so he has me over a barrel.
Yes, I think out of everything you have said this is the worst part for you, your left with the dogs, the bills the responsibility of the walking, feeding loving of the dogs. It's like many husbands who walk out and leave people with little ones, they are trapped in a life of responsibility and hardship, whilst the one that leaves is young, free and single again. If you are struggling possibilities are to sell up and rent, find a friend or advertise to share your flat with, or downsize/buy to rent, look into all possibilities.
What I would do is insist that he comes to take the dog for walks etc still, ok he can't take it to his houseshare room, but he can still drop round food and spend time with his dog, take it out on an evening when back from work, take the dog to his parents house for the weekend so that you can have some freedom to go out and meet other people too, no matter what the excuse he can surely find some time somewhere to be with his dog if he choses.
If he doesn't then to be honest, why do you not just cut ties completely, have him sign the dog over to you so that he can not just turn up and ask for the dog back if the mood takes him, and he is no longer in your life, if you could have a good relationship as friends the dog thing could work, but if not best to break ties, even though his dog walking money comes in handy, in time you'll both have new relationships and he is likely to just cut off anyway, I'd rather do it on my terms.
Get in touch with the PDSA with regards to vet bills etc and see if there are any services they can offer or think of to help you cope with two dogs whilst working and struggling, maybe the dogstrust too.

Make a point in talking to people on your dog walks, sometimes they can open avenues or even know people who are genuinely looking for dogs, or may offer to walk your dogs for you for free, you may place the dogs somewhere just by having a casual conversation. I wouldn't be calling a rescue at present, even as things are the dogs are better off with you, this time next year who knows what will happen, you might get a fab promotion, meet the man of your dreams and have a new house and life for you and your dogs, things change, hold on in there.