I'm going to voice what will probably be an unpopular opinion. I'll start by saying that I have always had rescue animals. Every last one has come with issues that I needed to address. None of them has fallen into the routine of my household without some give and take on both sides.
You made a commitment to that puppy 3 months ago that you and your family would love, provider, and protect her. Your family has not lived up to your promise and from what you've said you haven't made an effort to change your family's lifestyle to accommodate the dog. Shame on you all.
Your boys are 11 and 13; plenty old enough to think of the needs of someone else. It's about time that they found out that the world doesn't revolve around them. That there are times you have to do things you don't want to do because it'll benefit another creature.
Sit down with them and set a schedule of who does what on which days to meet your dog's needs. Walks can be family time. We use walks as a time to relax and chat about the day; to enjoy each other's company. It's not a chore. Of course when it's 10 below zero or raining I don't want to be out walking the dog either however I made a promise to my dog and I need to put his needs ahead of mine at times.
Before you pawn off your dog on someone else under the guise that she'll be happier maybe you all should actually make an effort to be a dog owner.