Yes, some shelters/rescues will let you 'try' a dog. So next time, look for that. We just recently got a new pup from a nice no-kill shelter (an 18 mos old purebred Sheltie). Their deal was that you pay the adoption fee and you have an agreed upon period to try the dog. If it doesn't work out, you bring him/her back and try another one. It was a great way to see if the dog you liked at the shelter would work at your home as they behave differently in your home! The first one we tried did NOT, and while she was so cute and sweet, she just wasn't for our family (too small and too nervous). We spent several hours at the shelter each time looking at all the dogs, watching them run around and interact with other visitors and other animals and really thought about how that dog would work with our household.
People can be so 'fussy' about dogs and dog behavior and trying to fix/dogproof/make it work with dogs. IMO, a dog works and fits in your family or it doesn't. Yes, I believe that truly any dog could be worked with to 'work' with a family, but at what cost!? Having to do so many things to make the dog work for you completely takes the enjoyment out of having a family pet, and causes so much stress. Yes, I do believe that having a pet is a commitment, but as I said before KIDS over pets every.single.time. I don't like how the shelter/rescue industry calls it 'adoptions' and really tries to play on our guilt and equate it to 'adopting' a child. It is NOT the same. End of story.
We've had a good dozen dogs in the last 20 years, and we've rehomed one as well. She was a Border Collie/Lab cross we owned from puppyhood to age 2. We'd had all variety of Border/Aussie/Heeler/Lab crosses in the past so I do know that they are high energy/need dogs, but this particular dog we had to rehome was not working for us. We'd trained/loved/worked/socialized her the same way we had our other successful dogs, but for some reason her 'personality' was different. She was strong willed and would behave for weeks, then run off on a bender and get into all kinds of trouble. She was much better off where she ended up rather than our being unhappy with her/she with us. We rehomed off of Craigslist too btw! (And I never got any nasty replies....but I live in a smaller town so maybe that's why??) We had several calls and we talked to the people, then took her out to the house that sounded like the best fit, and they took her to 'try' for a few days to see, and she fit great into their lives. We cheked up on her a few times over the next few months and they couldn't have been happier with her.
All our dogs have been 100% lenient to the kids. No growling (which I feel IS aggression in my world!). They will let the kids completely maul them--kiss them, sit on them, hug them, roll around with them, pet them until their fur practically falls off, whatever. They know they are to be lenient to the kids and we praise them and reward from the get go for that so they do it. They also know to just 'leave' if they are done and we will let them out. They don't need to growl...they just remove themselves from the situation. It is possible to find a dog that is great with kids...we've had plenty.
Before you get another dog, spend some time with people with dogs and see what kinds they have--size, breed, etc. and find out what they like/don't like about their dog/breed. Ask about past dogs they'd had and what they've liked/not liked about them. Really analyze your lifestyle and decide what kind would work for you. Though keeping in mind that breed types are just 'generalities' and a lot depends on a particular dog and how they were socialized and treated as a pup. But, you can get a good idea for what might work, and give you a starting point. So, when you go to the shelter you can say "We are looking for a dog 25-35 lbs, mellow personality, good with kids and cats, and doesn't need a lot of grooming." Or whatever you've decided. The shelter should then be able to steer you to a few different dogs they have that might work. Don't be afraid to leave without one! And don't get sucked into the cuteness! They are all adorable and deserving of a home....just not yours necessarily! Read up on a variety of sources for dog training (there are a ton of different ways...just like child rearing!!) and figure out what sounds like it will work for your family--also see if they say what breeds this might work well with. For example, Shelties and a lot of the 'herding/working' type dogs are praise/food oriented. This is an easy training method IMO and great for kids. They praise, praise, praise good behavior and give treats (who doesn't love that!) and basically 'ignore' the bad behavior, so the dog wants to only do the good and get the incentive. Just a small example, but you get the picture I hope!
Our new shelter pup was being rehomed because he was too high energy for the family that had him and they didn't like that he followed/herded the kids and raced around their small yard. This wasn't a bad thing for us! We live in the country and have tons of space, and we're familiar with the herd dog mentality. They need a 'job' to do and lots of exercise. We were looking for a dog for my high energy 11 yo that wasn't too big and that could keep up with him. We love the 'herding' instinct because they are so much fun to play/walk with and they stay right with the kids when they are out on the property running around. This one is just perfect for us. Within 24 hrs of having him, we knew it would work whereas with the other one within 24 hrs we knew it wouldn't.
So, sorry for such a long post that doesn't directly address your current problem...but in a round about way, I'm just saying, rehome the dog, and find one that works. No guilt. It's a dog. Not a kid. You'll all be better off in the end. Definitely check with the shelter, they most likely will take him back and rehome for you.