Outside of Seward's own writings, it's difficult to find any serious articles on the topic. But searching the phrase "Don't Worry Club" in newspaper databases turns up hundreds of references of a lighter sort. Sometimes these are just the little bits of humor newspapers of the time stuck between stories, like the one above. In one I found in The Brooklyn Daily Eagle, a woman complains that her husband's friends advised him to join the Club before getting married. Then there are longer humor pieces, such as "A Short Tale of the Don't Worry Club," which appeared in The New York Times in 1903.
But apparently there were also some semi-organized, or ad hoc, clubs. The one in Brooklyn seemed to be principally an excuse for going on Sunday outings. An article in The Eagle from July 1901 is entitled "Exalted Optimists' Outing." In this adventure, the "aggregation without a home, an organization or any hope of future temporal existence," traveled via the Long Island Railroad to Montauk. Reference is made to the baggage car and worrying while traveling through "no license" (dry) towns. I infer that they had a bar set up there.
Not surprisingly, it didn't take businesses long to exploit the doctrine's popular appeal. I've seen ads that mention the Club for no-worry loans, no-worry dresses, no-worry fur storage, etc. One popular use was on brass tokens. Tokens like these were already popular at the time and the Don't Worry theme was something all sorts of businesses wanted to be associated with. As the reverse of my "Old Prentice Whisky" token (above) illustrates, these made use of various good luck symbols, such as shamrocks, wishbones, horseshoes, and the then-innocent swastika.