If you need a luggage cart to take your bags back up to check-in, they're available ($3 IIRC) at bag claim. Then just walk to an elevator and go up to the departures/ticketing level.

If the connecting carrier's counter isn't open yet (you could walk by it en route from the arrivals gate to the escalators down to bag claim) then you can take your bags to Ken's Baggage on the bag claim level, and check them for $7 or $8 each.

You could then walk to the light rail station (other side of the garage) and take the train into Seattle ($3, 30-40 min.) From the International District station you could walk 4 min. to the big Uwajimaya complex - a great pan-Asian supermarket/food court affair, for a meal in the food court and a walk around; or else continue on the train to the Westlake station and walk 5 min. to the Pike Place Market for strolling and noshing. Spend an hour or two downtown, then back to the airport, fetch the bags, check in, fly off.