HI OP
I was in this position a year ago. Depending on where you live, if you have your notice to quit, take that to the council, that date will be the date you are considered homeless, you no longer have to wait for bailiffs.
Have a look at gumtree and spare room for private lets, also register with estate agents, you won't have a cat hells in chance in getting a let through them, but it will show the council willing.
Also if you have it, facebook local groups, if you have any pets, also post in those type groups as well.
When you go to the council, take with you your notice to quit, medical notes etc. You may find that you will spending a lot of time going to appointments at the council office.
It may be worth, looking into storage, if you are placed into emergency accomadation it might be really small (we were placed into a one bedroom flat from a 3 bed house)
With the council, don't be tempting to put a brave face on your disability, make sure you are open and honest about everything you can and can't do it might help with regards to getting you into a self contained flat rather than a B&B. With regards to B&B as you have a child if you are placed into one, they can only keep you in there for 68 (I think) days.
The only other thing I would say, is if you are accepted onto the list, bid on everything possible. I know you have said that your partner has problems with his knees, but if you are accepted for a property, then you will have to wait a year (each HA is different but its usually a year) before you can get onto a home swapper. So if you think you can cope with a 3rd floor flat for a year or more (might take a while to swap) then don't discount it straight off.
Nearly forgot, if you do get onto a housing list and moved into emergency accomadation be prepared to move in a short time. We had to move twice in the space of 6 weeks, once to the emergency flat and then into ours, but we were very lucky that we got a place very quickly