In order to starting doing business you'll need to file a Master Business License Application with the Washington Department of Revenue. You can find the application here: http://bls.dor.wa.gov/file.aspx
There's an option to register a trade name when you fill out the application. The application fee is $15, and its an additional $5 to register each trade name. Registering your trade name will help protect the name from being used by another business in Washington.
As far as limited liability protection, your business will enjoy the same limited liability protection as any other LLC, even if you're using a trade name or "doing business as" (DBA).
I like Gavin's answer. The default rule is that business entities (such as an LLC) protect the business owners from business liabilities. The exception to the rule of limited liability protection arises when the business owners are misusing the corporate form. One instance of misuse is intentionally making it difficult and confusing for those harmed by the business (including creditors) to bring suit against the real entity responsible.
I don't think your trying to make things confusing by using a different name on your sign. Nevertheless, the prudent thing to do is to register the name on the sign as a trade name (d/b/a) of the LLC.
Legal disclaimer: The answer provided above is for general information purposes only and should not be relied on. This answer does not form an attorney-client relationship. You should consult with an attorney of your choice to fully advise you about your legal rights and obligations