So we took one of our standard 100% cotton Hawaiian shirts and threw in in the wash with cold water. Although the instructions says to tumble dry cool, we didn't want to be waiting around forever for this to dry so we dried it on low heat. And since the shirt was a little wrinkled coming out of the dryer, we decided to just iron at regular heat so we could steam out the wrinkles. We wanted this test to be as real-world as possible.
We measured the length, chest width, and shoulder width before and after washing/drying/ironing. The only dimension that changed noticeably was the length. Before washing the shirt, the shirt had a top-to-bottom length of 31 inches. After washing the length measured 30 inches. This actually confirms the manufacturer's prediction that the shirt would shrink about an inch in length.
But what about the other dimensions? Well, to our surprise, both the chest and shoulder measurements did not change measurably. Again, this confirms what the manufacturer told us before hand.
We also did a subjective before and after fit test on one of our friends. He said the shirt felt pretty much the same before and after the test.
So that sums up this initial shrinkage test. We will follow this up with similar tests on other shirts from different manufacturers, including rayon shirts, and also different styles of dresses. For now, at least for this particular shirt, it appears that after washing, drying, and ironing, only the length of the shirt changes appreciably. So unless a 1-inch shrinkage in the length of the shirt is important to you, this Hawaiian shirt should fit you just about the same before and after you wash it.

Before Length = 31 inches

After Length = 30 inches
(1 inch shrinkage)

Before Chest = 46 inches

After Chest = 46 inches (no shrinkage)

Before Shoulders = 20 inches

After Shoulders = 20 inches (no shrinkage)