You probably can since this could legitimately be considered necessary for your daily life. In general, one is allowed to drive to and from work, school, religious services, the supermarket, and the like. If a person is moving and so they no longer live at one house and must therefore leave and go to their new house, presumably it would be necessary. However, you should use caution and still try to have someone else drive you. Just because you are legally allowed to drive for such a purpose does not mean that an officer will agree. You might have a great legal argument but you could still be arrested by an unsympathetic police officer.
This is not to be considered legal advice nor does an attorney-client relationship exist.
It is important to realize that there are two levels to this question: what the law actually is and what the cop thinks. It is small comfort to have the judge dismiss the case after you have spent the night in jail because the officer didn't understand the law. Having said that, there is one more consideration: the time of day. In my experience, very few officers are going to give you a hard time in the daytime. After dark, you should have specific answers in mind as to why you are behind the wheel, and (it perhaps goes without saying) do NOT have the smell of alcohol on your breath. In general, however, it is my experience that judges would agree that this driving falls within the rubric of "driving necessary to maintain livelihood" as specified by the statute.