Yes, you must sign the envelope! If you were to vote at a precinct on election day you would be required to sign a voter slip before you were given a ballot, the absentee envelope serves as the voter slip in this case. Envelopes that are not signed will not be counted, per state law.
Yes your ballot will remain secret. On election day the Absentee Board meets to conduct the process of ensuring that all the envelopes are signed, and opening only those that are. The envelope is then opened and immediately added to the stack of other ballots to be counted, the ballots themselves are not inspected and once a ballot has been placed on the stack there is no way to identify whose ballot is whose.