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Does a shuttle transport run between SeaTac and the Sheraton in Tacoma?
This hotel in Tacoma is no longer a Sheraton, it is now the Hotel Murano, you might check their website regarding a shuttle to SeaTac but it is highly unlikely. There is public transportation with Sound Transit so you might look at the Sound Transit web site for scheduled departures and fares.
There are a couple of options I know of from Sea Tac to Tacoma. Super Shuttle runs that route, and so does Capital Aeroporter. Those would probably be your best bet.