Cascade Rail Foundation has an extensive collection of these documents for the Pacific Coast Railroad (later part of the Burlington Northern), which the Milwaukee Road used as part of its transcontinental mainline between Maple Valley, WA and Black River (south of Seattle).
Knowing that the 5057 was originally number 388 when it was built and delivered to the Milwaukee Road during the summer of 1965, we went through the dispatcher train sheets for that period. The earliest sheet we've found that shows the 388 on a train was from July 31, 1965. An excerpt can be seen below, with the entry for the 388 highlighted in orange.
|Click to enlarge.|
Unless new information becomes available, we consider this to be the first trip of the 5057 to Washington.