Yes, some private parks will "ban" your dogs. Just go elsewhere.
Most public campgrounds do not have breed restrictions and do not "ban" dogs.
When you are camping ALWAYS follow the rules. Keep your dogs on leashes. Don't leave them unattended. Clean up after them.
Remember that if something happens, no matter who is at fault, your dog(s) will be looked at as the problem by default because of their breed - so you want to do everything to prevent any complaints.
Don't throw an attitude. If you get questioned, always be understanding, take responsibility, and be honest. If something does happen, don't get defensive. Listen to the ranger/authorities, accept responsibility for actions that you or your dog did (e.g., I did drop the leash accidentally or I didn't see them until they came around that corner on the trail, etc.)
Don't say "my dog wouldn't hurt a fly", etc. Those types of comments are overused to the point of being meaningless. Be a bit reserved and minimize contact with strangers - who may do something that scares your dog and cause a reaction that your dog will then be in trouble for. Remember it's better to be distant from other campers than it is to risk their doing something that will get blamed on your dogs.
ALWAYS carry copies of their rabies certificates (not just the tag). I keep a copy in my motorhome, my trailer and my SUV. Having that certificate on hand can prevent your dog from being impounded. And it shows that you are a responsible owner.
If other dogs are running loose, being aggressive, etc., be vocal in telling your dogs "what a good dog. Just ignore that bad dog" and "please get your dog and keep it away from mine", etc. You want other campers to notice YOUR appropriate behavior and the other dog's (and owner's) inappropriate behavior. That will help protect your dogs if another dog starts a fight, etc.
Again, always keep your dogs on a leash. As long as your dog is on a leash and you are holding the leash, you have minimized (if not eliminated) any cause to blame your dog(s) for any incident that could happen.
For the younger dog, just work with her while you are camping. Most people will be fine with your dogs if they see that you are attentive to their behaviors, making an effort to instill and promote good behaviors, and that YOU are not being rude, aggressive, or snotty to others about your dogs.
I have only had one person go through my campsite while my dogs were there. It was a yound gir! about 7. I was sitting in my chair and holding my dogs' double leash. The girl came running through my site and ran straight towards my dogs. When my Bat-dog started barking at her, she froze. She was definitely startled and scared (the girl). I told her to turn around and go back and then go around the campsite. She did and there was no other problem. That trip, there were a good dozen kids hanging out just below my campsite at the creek. They were all extremely great kids and their parents were too - we just agreed to reasonable boundary markers so the dogs could adjust to their coming and going without feeling their "territory" was being infiltrated.
Bat-dog has people fear issues and as long as I follow the above guidelines, I have never had a problem with her and other campers. Everyone is very understanding. But I'm honest and will explain her issues if I see someone looking sideways at us. The rangers are always very nice about her. I have even had some make an extra effort with her to help her learn people are nice.
Always keep your dogs on a leash. As long as you have them on a 6ft leash and are holding it, you can prevent most bad situations. Those you can't prevent, you can show that you were always in control of your dogs. This is the most effective way to protect your dogs.