You definetly have your hands full. BUT.. what if you ask these people who say that the babies are theirs, to bring the Mama dog with them. If these babies are as little as you say... they (the owners) should have the Mama dog. And she would know her pups. Ask the questions:
Why would they have left them in a shopping cart??!!
Where's the Mama dog?
What are they doing with three puppies??
Good luck to you. I wouldn't just hand them over to just anyone. I say finders keepers, and you can find GOOD homes for them. I wouldn't really want to give them over to people who left them in the first place. You have a hard job to do. Hang in there. I know you will do what's best. My thoughts are with you.