紧急求助：不签字的OFFER有效吗？

I have just received a offer from a private university by email. The offer was sent by HR with the name of Employment Administrator. In her email, she said it's official offer. However, she also asked me to sign a release letter for background check. Usually, the background check should be performed before sending an offer, and the offer should be mailed and signed by the head of the organization. The offer did not mention the start date.

I signed and faxed the offer letter back as they required. Is this an valid offer? Am I responsible for my signture?