Am I legally bound to give 60 days notice to move out if I am a month-to-month tenant?

If you are a month-to-month tenant, the term of the lease is only 30 days. That is, at any given time, both you and the landlord are only bound for 30 days (one month). Each month can effectively be thought of as a new lease term. A consequence of this is that if there is no lease, as a general matter, either party can terminate the tenancy on only 30 days notice. You should check back with the expired lease though, as well as any communications regarding it. Even if the lease expired, if you and the landlord had correspondence clearly indicating that you would renew under the same terms, that might be considered to be a renewal of the lease and you could be bound to it. Also, it's not impossible the lease had terms that would affect even after it's expiration, if, for example, it said, "upon expiration of this lease, if not renewed, the tenant shall have a right to hold over subject to 60 days notice of the termination of tenancy"--contracts (which is what a lease is) can have terms that survive the expiration of the bulk of the contract.

However, if neither of the above applies, then you'd need 30 days  notice of terminating  your tenancy.