I am just utterly, utterly
that you could consider rehoming a thirteen year old Staffy that you have had since a puppy. Hats off to Dooin, for giving you some very constructive advice. If you didn't know already, most rescues are awash with Staffies and Staffie Crosses, and many, many healthy, young dogs are put down each week as no homes can be found for them. An elderly staffie is really going to struggle - it will be a miracle if they can find a place in rescue.
To help the rescue, make sure you have gathered together all their records, such as vaccination cards, make a note of when they were last wormed, and put together notes on things like their health, any medication, behaviour, training, any special quirks etc. Are you expecting the two dogs to be rehomed together, or do you think they will be split?
Can you not look again at persuading landlord to let you stay? How long are you leaving poo in the garden? I really can't see why you are leaving poo out there? You wouldn't do this when you are walking them. Do the dogs stay out there unattended? If so, they are at risk of being stolen and could be causing a nuisance to your neighbours. Have you been able to talk to your neighbours about the dogs, to explore their concerns and see how you can resolve them?