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Unless you legally adopt the children, which would require the biological fathers to either agree or be found unfit by the state, and would cut off all future obligations by them, they will continue to be liable for supporting their children, not you...But it's not always as simple as that.
The primary law that directly addresses stepparent liabilities is RCW 26.16.205 (http://apps.leg.wa.gov/rcw/default.aspx?cite=26.16.205). The good thing is that your liability ends when the marriage ends, so you shouldn't be ordered to pay child support after a dissolution. But that statute also says that "expenses of the family" are "chargeable upon the property of both spouses...and they may be sued jointly." So while you're married, it's conceivable that you could be responsible for damage caused by one of the children (see http://apps.leg.wa.gov/RCW/default.aspx?cite=4.24.190). Or you could find yourself paying for some element of the child's care after divorce if it was put on a joint credit card, because creditors aren't bound by family court decisions (Visa doesn't care whose kids you paid expenses for, you have a contract with them), or even coming after your income for a debt she incurs for the children because community property laws say that anything earned during the marriage belongs to both spouses.
It's also possible that the marriage could lower the biological fathers' obligations, in effect making you liable for some support. As a general matter, a new spouse's income is not a reason to change child support, but if there are other grounds for deviation from the standard calculation, it can be one of the factors taken into consideration.
You could talk with an attorney about any fact-specific concerns you have, and consider whether a prenuptial agreement might help keep things on the right track (typically matters involving children can't be controlled by contract, but you can come to an agreement about keeping finances separate during the marriage which may help assuage some of your fears)...but it's impossible to cover all contingencies. Hope this helps a little!
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