Cotton Lawn is a thin fabric, but typically a high density, and thread count, sometimes having a sheen . Think of high thread count sheets .. fine and dense threads. Do use a very small needle in your machine( 65 or 70). High thread count fabrics can have holes left if you need to take out some stitches and you use a large needle.
I have not given the feel test of the Connecting Threads cotton lawn fabrics. It would have been helpful if they gave the actual thread count so the consumer can have some guage prior to purchase.