Yeah, in the movie and series Adams was falsely accused of murder and had to go into hiding. The real John "Grizzly" Adams left his wife & children with her relatives in Massachusetts and went to California to trap & train wild animals for circuses.
In all fairness, the guy had lost just about every penny he had when a load of shoes he had planned to sell to the people heading West were destroyed in a fire. His family would have been pretty well screwed, anyway. They were likely better off than they would have been had he taken them with him.
As a seven-year-old kid, I didn't know any of that, of course. I just knew I liked the show.