I looked it up and that one does have a shutoff. It's an electric solenoid behind the dash. If you turn off the heat, it should shut off the water.
Having the core replaced might get pretty pricey depending on what all they have to remove to get at it. This website says anywhere between $300 and $1000.
HEATER CORE

edited to add: I just noticed that even though I googled 2005, that site is for a 2002. It might not apply.

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― George Orwell

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