I'm not sure where the idea of using foreign addresses and large envelopes comes from. With the Spectator, for example, I trim the page down the line to the left of the across clues, along the bottom where it just edges the end of my address and up the right-hand edge of the grid. This produces a piece of paper that fits nicely into a 7"x 5" envelope, which is the sort I always use for crosswords. I've won the Speccie or got a runner-up prize four times in the past couple of years with just such an entry.
I use exactly the same approach for all and have had multiple wins on the Mephisto, the Azed, the Guardian Prize and so on. At one point, I had so many dictionaries incoming that I nearly ran out of good homes for them!
It appears to make no difference whether one submits early or late. I cannot estimate the odds, but - whatever they are - they don't seem to have been working against ME. My advice, therefore, is...Keep sendin' 'em in!!