This is a double edged sword.
I was in the same situation couple of weeks back. My attorney advised me that I should immediately change address with BCIS as post office does not forward government mails. Therefore, it is very important that BCIS has your correct address.
Now, I was left with three choices, one don't move at all (which was not possible). Second maintain both of the apartments (this was also not financially viable), and third change address with BCIS and pray to God that things don't fall apart.
So, I opted for the most logically appealing option, that is, I moved to new apartment on Sept 7th and sent the change of address request to BCIS via my lawyer immediately. I also maintained the first apartment for additional 15 days just to make sure that in case there was a mail that left BCIS office, before the address change is updated in their systems, should not return.
Now, I've surrendered the old apartment today and I'm just praying to God that nothing should go wrong. As there is no way for me to know that BCIS actually updated my address in their systems or not.
I guess you'll need to take a careful decision on handling this situation.
Thanks