First of all, change your address with the post office. They will forward your mails to your new address. You said, they don't forward DV letters, it might not be true because they are your mails which are addressed in your name and your address. Still, make sure with the post office. If so, hold your mails and pick up from the post office yourselves. 

Sometimes, even you change your address, post office don't deliver every letters. In my case, I changed my address but they delivered some to my new address and they delivered some to my old address. So, you might wanna keep the mailbox key of your old apartment. Normally, the residential properties do not lease your old apartment for at least few months because it takes time to clean and they sell other available ones first. May, June and July; just three months go and check your old mailbox regularly when you will be expecting your DV letter.