Welcome to OFF!
Lake Billy Chinook is partly Native American Tribal land. You can be in hot water in more ways than one, if you fail to know the reg's on, or around, that body of water. I'd go anywhere else but there.
According to the 2011 ODFW reg book, you do NOT need either an angling (fishing) license nor a shellfish license, to catch crawdads (called crayfish in the book). Both of the methods that you mentioned to catch them are legal (by hand, or in a trap). I see no other restrictions other than you cannot use hooks to catch them.