I have done a lot of research online on this topic and am familiar with the responsibilities when someone has a clearance, but I have not been able to find any clear answers on if those responsibilities are still applicable if your clearance has become inactive or expired. I have asked my FSO, but they were not particularly helpful.
It goes without saying, the NDA you signed when you received your clearance, does not expire when your clearance is not renewed or you no longer have a need to know. When you leave your job you will be expected to be debriefed, and you should be notified of your expectations with regards to the information you know, when that happens.
Once you no longer have a need to know, your clearance should immediately be revoked, you are no longer required to notify your security manager (FSO) of your travel. I will go ahead and repeat the fact, even though you are no longer required to report your travel, you are not allowed to disclose anything you are privy to.
I intend to do a lot of international travel, and I don't want to wake up one morning to find a Blackhawk hovering outside my house and being asked why I hadn't told anyone I went to Ireland.
This does not happen even if you hold a security clearance. If you do international travel without notifying your FSO, then that security incident would have to be reported, and your clearance could be impacted. Since you no longer plan on having a security clearance that requirement is no longer required.
What are the responsibilities of someone who possessed a security clearance but is no longer employed by the DoD?
You are required never to reveal any information that is classified. You are not required to report your international travel, since you will be self-employed and I assume you will NOT hold a security clearance once you leave the DoD, you wouldn't have anyone to report the travel to anyways.
Once you have actually given up your security clearance, you will be told what expectations if any there will be, including reporting contact with foreign government representatives. At the end of the day, any requirements after you leave the DoD, will really depend on the amount of knowledge that you have been privy to. The best person to ask specific questions, with regards to your specific knowledge, really is your Facilities Security Officer (FSO).
With regards to a CAC (Common Access Card), there are lots of DoD employees with a CAC, that do not have a security clearance.