Personally, I wouldn’t bring them. The odds of being caught with them might be low, but if something did happen, say you were forced to use your firearm to defend yourself, the legal stakes, especially in Seattle, could be very high.
Get some inexpensive 10 rounders. The capacity is more than adequate for any situation you could realistically expect to encounter and reloads can be immediately available.
As far as seeing the Pacific, it will be a full day at a minimum. The easiest way would be to drive south of the Olympics out to Ocean Shores and drive on the beach. The more scenic way, by FAR, would be to take ferry across the sound, drive US 101 north of the Olympics out to Rialto beach and take the short hike to Hole in the Wall, which is one of those iconic Washington coast sights. If the weather is good, Rialto would be my pick.