Kateryna, you are so not a failure. Don't get down on yourself like that please. We've all been in your shoes. I do believe right now you are just overwhelmed.
LP will help with whatever she can, we all will. I also have a dog who at 7 years of age I seriously considered either PTS or giving her away. The only person I would have given her to was a behaviorist who was willing to take her. I couldn't fathom ever passing her off to somebody else. But you never know until you meet some of these rescue people if you would trust them with your furbaby. I would have given my girl to this one man. But nobody else. Explore all your options, and if you decide at the end that putting your pup to sleep is the best option then at least you will know that you did all you could. Seeing your vet is a good first step. You my dear have a very full plate as it is, and it's already been said, but at this point you must protect your twins first. Don't beat yourself up for this and don't berate yourself. Be strong, you know you are, you have twins for cryin out loud (no pun intended).