I've been through the lottery, and my experience leads me to suggest (especially based on your timeline): don't waste your time. It takes a good chunk of the morning and even if you are selected (1 in 25, if you're lucky), the permit is for the following day. The number of permits they draw is not a fixed number, but depends on the number of people post permit... Only 10 (I think) per day, so if one of the numbers they draw has 10 people, gameover.
If it's swirly red interbedded sandstone you're into though, check out the Coyote Buttes area. It's the same formation as The Wave and 0 permit restrictions. There's a convienent campsite right there at the AZ/UT border, too.
Yey! Now that's what I call useful advice. How far is it from the main road to coyote buttes? Google places a quite random flag just south of the wave...
Although sadly it seems you now need a permit for there too :(
OK, so to be more specific: Stateline Campground
, and the Wire Pass Trailhead
(~one mile away) does "require a permit", but if you have a federal land access pass, it is either included or $6 at a self-serve pay station. I went in May and set up camp in the early afternoon then hiked the trail starting at about 3pm and saw no one else the whole time. The trail goes through the same formation as the Wave and is amazing: