OP, your vet should be your first port of call. Even the sweetest tempered dog can be a bit snappy if they have something wrong with them, which may not be apparent - I'm thinking particularly of toothache/dental problems or an ear infection. These can be excruciating and not easily visible to you.
In the mean time, please reinforce with your children good "dog manners", so no hassling, poking, prodding and ensuring your dog has a safe place to go where she can rest undisturbed.
Your vet can probably recommend a behaviourist and it's worth doing some work with them if this is indicated.
Having her PTS isn't necessary - even if she does need to be rehomed at a later date there are many adult only homes who adopt (we are a good example
) as one of ours is nervous and timid around children he doesn't know.
If you do decide to rehome her, please, please work with a reputable rescue - they can ensure the dog is given every possible chance at being carefully rehomed in the right environment, with the support and backup needed. One of the benefits of rehoming via a rescue is that usually a careful assessment is made of the dog's character, and good rescues will provide lifelong support and behavioural advice every step of the way.
The fact that she is a Staffie X doesn't mean she is any more dangerous than any other dog - quite the reverse. Staffies are known as the "nanny dog", and I've met some absolutely lovely ones.