Acrobat 9 and later do not allow changing a document after it has been signed unless specified in a Certified document. All signatures must have a field prior to being signed. You could have an additional field for the application of the dates effective and then apply the last signature I guess.
Acrobat versions prior to 9 had a bug which allowed you to do what you did. In Acrobat 9 this bug was fixed. In order to do what you want you need to certify (apply special certification digital signature with your certificate -- it is "Certify (Visible/Invisible)" choice in the "Work with Certificates") your document first. In the process of certification you can specify which modification permissions you allow the users of your document to make after signing. Certification signature must be the first signature in the document.
Thanks, Zildjian and isakten. You pointed me in the right direction. What I ended up doing was just placing a text form field (for the Effective Date) and then the needed signature fields. After everyone has signed, I can input the Effective Date and save the document and I don't have to sign it myself or apply any kind of certification. I just have to remember to put the text form field in before I route the documents. This is the Correct Answer, at least for my situation.