How quickly does parking fill on the Stroll Saturday? Are there any Park/Shuttle options?

Although we have a home on the Cape, I am not familiar with the stroll you're referring to. But in general, traffic in / around Chatham is very seasonal. From about June 15 to September 1, there often are more people than there are facilities for them, so parking during that period might prove difficult (esp. in July & early August). In January thru March you can probably bowl down Main St. and not hit anybody. The best times to be in Chatham (IMHO) are the two shoulder seasons May-June and Sept-Oct. when the weather is still pleasant and most of the tourists are gone. Since the town tries to accommodate that July crush, parking (and other accommodations) tend to be more than adequate outside that peak season.
I think she was refering to the Christmas Stroll that does bring in a large number of people for that weekend...which was Dec 13th - the 15th this year.