You seem to have an accurate grasp on the alternatives. Yes, you can discharge your personal liability for the debt in bankruptcy. Be aware, however, that any lien on any property recorded by the county in connection with the application for county assistance will not be avoided by the bankruptcy. This may not be a problem as counties don't seem interested in rights in personal property (such as cars, etc.) but the lien could attach to future real property purchased. Before you get any farther into the county process, you should consult with an experienced bankruptcy attorney. Be sure any attorney you are considering retaining is fully familiar with the difficult relationship between the county aid and bankruptcy and the unavailability of bankruptcy relief for statutory liens. Choose an attorney with considerable experience in Chapter 13 ("in how many cases in the last year have you achieved confirmation of a plan for your client?") and ask whether Chapter 13 can effectgivbely stave off county demands.
Best wishes for a favorable result, and please remember to designate a best answer.
This answer is offered as a public service for general information only and may not be relied upon as legal advice.