We stayed at the Warwick in Seattle and there was a shuttle from seatac.. I am sure that there is one to the Sheraton... if you call the hotel directly, they should be able to give you specific information... enjoy your trip!
There is a shuttle from the Sheraton to the airport. It is the Grayline shuttle, it runs about every 30 minutes and is only about $8.50. I have used this shuttle many times and the hotel can give you full details if you call them.