What to do with a dog that we love but can't keep?

We've had Lucy for a month but shortly after we got her my wife was diagnosed with SEVERE hypertension. The doctor basically said that my wife is one chewed shoe away from mive heart issues. Not that Lucy chews it's just that she occasionaly gets into a bit of trouble, a typical dog. Without the hypertension issues there would be no problem. We've tried all the local shelters and the humane society but they are all full and we won't take her to a facility that euthenizes. Are we missing any avenues that could help us?