How can I establish paternity after father has passed and been cremated in order to receive survivors benefits

The Arizona Bar Association has free help available if you follow this link scroll down to "FIND FREE LEGAL HELP" so that you can choose your County.  I was hired in this fashion for a similar case in Nebraska through the Volunteer Lawyer Project.  Yours is not an easy case and you need help. In the case I handled, the father had been cremated and buried in Mexico.  Depositions of the mother and grandmother were taken to establish identity and paternity.  I also had the further complication that the name did not match the Social Security Number.  However, we were successful in the end. If you have been turned down twice, you only have sixty days to appeal, that is, sixty days after the date of the decision.  Do this immediately whether you have located a lawyer or not to preserve your rights.  You can do this in person or by phone through your local office.