I keep a cat, despite being allergic, like Vicar's ds I find that over time I become more tolerant of them. My cat moves out regularly, I find my allergy is actually worse when he is not living in the house, simply sitting near someone who owns a cat can be enough to start the itching and awful dry eyes and wheezing. When the cat is home I use Piriton for the first few weeks, limit my contact with him and slowly start seeing improvements.
It does sound you like have things tough atm, but I can't see that rehoming the dogs will make things an awful lot easier for you. How much hard work can two 13 yo dogs be? You'd still need a strict cleaning routine, to keep on top of dust, as Vicar pointed out.
I can't see how active they can be at night? I have a collie x pup. He is active at night if he has not had enough exercise, but even for him throwing a tennis ball up and down the field for 45 minutes an evening is enough to keep him mainly under control. Surely 13yo dogs would need little more than a gentle stroll twice a day?
What support do you have? Is there anyway you could access more support? As you have a sick child there are charities that might help you keep your dogs, PAPAS might be able to offer some support around the dogs and I would look at finding local asthma support groups if you have not already done so. Maybe ask at your sons school if there are any other organisations that could help you.
Rehoming dogs of that age will not be easy, nor is it the best thing you could do for them, particularly at their age, they will find it extremely hard.
If you genuinely feel you have explored all other options I second whoever said to contact Cinnamon Trust, you could also try The Oldies Club, but you will have a wait. Unfortunately rescues are full to bursting with healthy, trained young dogs, older dogs are often over looked because of this.
If you can find anyway possible to keep the dogs in your home, despite being left longer than is ideal, they'd have a better life than the alternative.
Good luck and I hope you can manage to access the support you need.