You need legal advice. See if you qualify for assistance. In general, the notice should have given you a date by which to leave. If you do not leave then the landlord has to file the eviction complaint. After you answer it will take a bit of time for the hearing to be scheduled. When you appear and give your reasons, the court will often give you time to find a new place. In my state, this can even be as long as several months. The important question will be your ability to keep paying something towards the rent.
Once you have been served with the summons and complaint of unlawful detainer in King County, the sheriff likely will be at your door in about 3 weeks from the date of service to remove you and your children from the property unless you win at the show cause hearing.
The facts that you have 2 minor children, have bad health, or are low-income are irrelevant if the basis for the eviction is that you are not paying rent. The court has no power to delay the eviction for you if you are behind in rent and the landlord has complied with statutory duties. The unfortunate fact is that the landlord still has to pay the mortgage, taxes, and other expenses.
If you have a definite plan to move out but just need a few more days, the landlord may be willing to give you the few days in exchange for your promise not to cause damage to the property and leave on or before the agreed time.
There are volunteers located in the superior courthouse who help tenants facing eviction. The volunteers can help you determine whether the landlord is in compliance with the statutes. If the landlord is not in compliance, that may be a basis for you to win at the show cause hearing. Even if the landlord is in full compliance, the volunteers may be able to help you negotiate with the landlord in giving you a few extra days to move.
You can also use WA LawHelp to locate free or low fee legal help: http://www.washingtonlawhelp.org/WA/index.cfm .
You need to act quickly. As indicated on the summons, you have only a few days to act.
If the eviction is based on non payment you have to be prepared to pay rent. You can, if you are disabled, request "reasonable accommodation" under the ADA and the WLAD. If there is an eviction hearing scheduled, the Housing Justice Project has volunteer attorneys at the courthouse who can review the pleadings and see if you have a defense. Sometimes defenses are pretty technical, and an attorney may spot an issue that you can't.
But the bottom line is if the assertion is that rent has not been paid, and you cannot bring it current, making other housing arrangements is in your best interests. If you move out completely before the hearing and tell your landlord you are out and give them back the keys so they know you are out, the Eviction court loses jurisdiction to grant relief to your landlord at the hearing because possession is no longer at issue. They can amend their pleadings and sue you for unpaid rent but they cannot get a writ and you won't have (with any luck) a final order showing you were evicted.
Hope this helps.