My kids are "part Cherokee" but I'm scared to take them to meet Native youth groups.
I appreciate the questions you are asking yourself and the great care you are taking to consider what the best steps for your children are. It sounds to me like you have a sense of what you might like to do by your saying, I feel like if they want to really dive into being Cherokee and find out what that means, that's good. I agree.
I also believe that this would be beneficial for every American to do, not just based on whether they are descendants of American Indian/Alaska Native people. Gaining knowledge and experience of other people's cultures and lifestyles is incredibly important for us as individuals and as a society. It's important because it teaches us how to better relate, understand, and empathize with the people we share this world with.
For the second part of your question, will the majority of Native people be offended if my kids show up and say "I'm part Cherokee"? You're right in acknowledging that this can't be answered the same for everyone. I think testing the waters might be a good approach to take. No need to go out and proclaim your kids heritage by buying your kids a buckskin pow-wow outfit and dancing around a bonfire at night :) If you decide to take your kids to different youth or pow-wow events, join as an observer making sure to be respectful of the activities. Often being a good observer will give you a sense of what level your participation might be welcomed.
Another good tip is to take the organizer's lead. Sometimes the M.C. will provide opportunities for non-natives to participate and get involved in some dances. Most of these events are public and are put on as community events to engage both Western and Tribal communities in order to educate and provide opportunities for learning and relationship building. Try them out. When you talk to folks, you can say something as simple as, "We've always been interested in learning more about Native American history and we're happy to be here now."
When and if you feel comfortable sharing your children's ancestry, do so in a way that is respectful and light. Then feel the person out. If they don't respond, maybe leave it there and change the subject as you would with any other new person and topic. It's common for Natives to think about a topic then bring it up again at a later time, usually in a passive way… so keep showing up to these events and see how the relationships you build may help your children, and yourself, learn more about where their ancestors came from.Happy Searching!