If you sign a money order made in your name then anyone in your family with a checking or savings account should be able to deposit it in their account, and then give you the cash (either out of pocket or from the available balance in the account, if sufficient funds are available). If you already have an account in your own name then you should be able to do the same. I did this for years with my mom's disability checks. She signed them. I deposited them in my account via the ATM, and then withdrew an equal amount in cash (my account had more than enough available balance so that I didn't have to wait for the check to clear). I never had any problems at all doing it this way. As far as I know a check or money order endorsed by the payee is as good as cash and can be deposited in anyone's account.
BTW, sometimes school photo ID is accepted as proof of identity. I'll second the idea of getting a state ID card if you don't plan to get a driver's license. I've had mine since 1988. Very useful for many things, and as good as a driver's license for ID purposes.