I completely understand where you're coming from as I was in your situation a couple of years ago. I have three children and at the time they were all under 4. Our youngest had just come home (international adoption) and I was completely overwhelmed and exhausted! I know some people can successfully juggle many kids and dogs but I am not one of them either.
I gave up one of my dogs to a rescue group several months after our dd came home. Now, if I could go back and do it all over I would NEVER make that same decision but I'm not about to tell you that you're an awful person for considering it.
I also know as a stay at home mom that when I have a few minutes of peace, and those are rare, the last thing I want to do is spend it exercising or training a dog but it's just one of those things that has to be done.
I don't really have a lot of advice. Looking back I now see if I would have just given it more time, maybe six months, everything would have worked out and I would still have my dog whom I miss very much. So if I were you I would try to give it more time. Your infant can't be very old and it has always taken me 6-12 months to adjust to a new baby. Get back into obedience classes. You'll have fun with the dog and get a break from the chaos that often comes with little children.
Do you have a dog park nearby? We fostered a wild & crazy active dog last summer and the only time I could tire her out was to take her to the dog park. Oh, and that was a lot of fun too! I would leave the kids with my husband after supper and take off.
But, if you have already made up your mind to let him go, you're still not a terrible person imho. You haven't banned him to the backyard or dumped him at the nearest pound/shelter like far too many people do. You are trying to do your best by him. Find a good rescue group and let them help you find a family for him.
More advice, for what it's worth...if you get another dog sometime in the future when your children are a little older (and no, I'm not one who thinks you should never have another dog) get an adult. Both of the dogs I have now came to us as adults and they are sooo much easier than puppies when there are young children in the home.
Good luck and don't feel insulted by all the posts. None of us have walked in your shoes and you have to do what's best for your family and your dog.