Well, first to this:

Turns out it is someone who he showed the apartment to a couple of weeks ago and she expressed an interest in renting it. By the way he apparently was showing my apartment to people without my knowledge while I was at work. Obviously he must think he can do that without me being home.
I can find no statute controlling the landlords entry so guess what that means? He can enter without notice to you with you there or not unless there is a local law or it is in your lease.

So, now that that is out of the way;

If you get anymore mail to your residence for the new tenant, I would simply marked it "adressee unknown return to sender" You do not have to place post office for the new tenant.

Then, I would call the power company back and in form them you are still the tenant and you have made no request for disconnect. I would not even add that you will be calling them by the end of the month. That is irrelevent. If they do disconnect prior to your request, you need to file a complaint with the public service commission. Be sure you have the names of the people you talk to at the power company.

If the person you speak with refuses to acknowledge you have no intention of turning off the power prior to the end of the month, speak with a higher up and do so until somebody gets it through their thick head you are their customer at that address until you give notice it has changed.

Curious as to what the LL said when you assured him you would be occupying the place until the first of the month.