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.
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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.
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