As a qualified pet behaviour counsellor and someone who has worked in various rescue centres for a total of 20 years, I usually stay away from this topic. However, I have read this and OP, I think you are being very harshly treated.
I am always amazed at the naivety on here. "No kill rescues" as they are called on here are a terrible idea. The only way they could be justified is if the tap is turned off at the other end; breeding of unwanted puppies. I have seen dogs languish in kennels for years, leading a miserable existence with no hope of being re-homed (realistically). Requirements might be 'experienced owners needed with no children, no other pets, no visitors, own land/very large garden. Cannot be exercised off lead (currently), no contact with other dogs etc.' There are many dogs like this in rescue centres. Such homes are vanishingly rare. And when they do appear, why would they want to take such a dog when they could have a wonderful time with a sociable dog, long walks with friends etc.?
At the other end, having worked in a Local Authority stray dog kennel, we had to select dogs to PTS because they had done their time (7 days) without being claimed and had no chance of being moved to a re-homing centre as they had no space. D'you see my point? I will never forget those days, taking delightful, friendly, sociable, often young and healthy dogs to the vets. those of you promoting rescue centres with non-destruction policies need to get out there and look at the grim realities of this country's dog population.
To get back to the OP; there are thousands of lurchers in rescue, many of which are confident and sociable, without homes. I am not surprised you are struggling to find one to take her. It is devastatingly sad for your family and the dog but I applaud you for taking responsibility. To those of you giving OP a hard time; you are doing dogs no favours. There is a fate worse than death for dogs, I've seen it. If only more owners took responsibility for their dogs and didn't leave it for rescue centre staff to do (or fail to do) days/weeks/months or years down the line.
Most people will not/cannot train their dog so the constant pleas for people to do that on here (especially when it involves children and aggressive (for whatever reason) dogs) are a waste of keyboard use. OP will have thought of that and discounted it, I guarantee it. It obviously is not an option for her. I have a young family and two nervous aggressive border collies; I know how bloody hard and relentless it is.