The fact that you do not have a mail box, does not mean much, since you do have a physical address, that the fire department would come to, should that house catch on fire or there was a medical emergency there. The post office, the county/city planner, utilities, county/city tax people and your builder/banker/mortgage company, all know what the address is, even if it's just a plot of land without a house on it yet. You DO NOT want to have too many incorrect addresses listed as your physical address. I have a friend that always gets denied the first couple of times when he buys firearms because of him having several different physical addressed "on file" of which he never has actually lived at, but were places he had mail sent when he was "between" addresses.
I live in Washington state, I do not get mail at my physical address, but rather at a UPS store that can sign for mail and packages that need to be signed for. When I applied for my CPL, I asked what address should I put on the application, my mailing address or the address of the place that I actually live (sleep, eat, call home)?
I was told that it needed to be the actual physical address where I have utilities, pay rent or mortgage for the CPL application, and that should ALSO be the address connected to my drivers licenses. And if it was not I needed to correct the physical address on the drivers licenses (which was done online), and then I was to put a sticker on the back of my DL to show the corrected address. Which for me made perfect sense, because I live in one county and my mailing address is in another county, and I could ONLY apply in the county that I actually lived in to get the CPL.
As for buying a gun without a background check, if you are buying from a FFL. pawn shop, gun shop, gun smith, you will be having a background check done, and you WILL be filling out a form 4473, and you had better be filling it out with CORRECTED information, even if you do not like the idea of putting down your correct name(s) and or address, you best be doing so.
The actual form is below, print it out and read it over for the information that is asked for.