You have asked multiple questions, and your questions raise a number of issues. While I cannot opine as to Washington law, and recommend you contact a Washington-licensed attorney to finalize your dissolution, I advise as follows:
First, a business bankruptcy is certainly not a "must." Unless you intend to reorganize the business such that creditors will all receive some portion of the obligations owed to them, it is unlikely there is any great advantage to a business bankruptcy.
Second, if you have personal liability on account of guarantees (letter of credit, loan, real estate lease, etc.), then you my find personal bankruptcy an appropriate option. With hope, you have no or limited personal guarantees.
Third, you ought formally dissolve your LLC by following state statute. Typically a notice of some type is published and/or provided to all creditors. Depending on the state, you often file Articles of Dissolution after some waiting period. As state law varies on how and when the dissolution is complete, you should consult with a local attorney.
Fourth, you will need to follow state law in distributing any assets. You indicate there are none. If assets were available, they MUST be used to satisfy creditors before being distributed to members.
Fifth, you should file a file tax return. Discuss this step with an accountant familiar with business and LLC tax matters.
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