Must my child guaranty payment to the nursing home on my behalf?

No. It is illegal to require anyone, including a child or spouse, to provide a payment guaranty as a condition for admission. 

A child or spouse can be asked to be a “Responsible Party” requiring that they assist in filing for Medicaid and other matters.  

Do not sign an admissions agreement without completely understanding it. Avoid signing documents while at the nursing facility - there is no requirement for you to sign them on the spot. Read more about this here.