Some good advice here. I've drawn the same unit the last 2 years in Wyoming. There is plenty of public land. The first year, I hunted about a week to 10 days after the opener and hardly saw anybody at all. Saw decent numbers of animals on public but the biggest one I saw was 13" and I shot it. Last year when I hunted, I got there 2 days before the opener to scout. The biggest buck I found stayed in the same spot for 2 days and I shot it less than 10 minutes after legal light on opening morning. Not what I'd say big but it was over 14" with good mass and a big curl. A neat buck and I was happy. There were a pile of hunters roaming around when I left.
As others have said, it depends on what you want out of the hunt. Chances are good if you went out before the opener and scouted, you'd get one good buck on opening morning and then you'd be dealing with a lot of hunters to get your other buck. If you go later, there probably won't be much for good bucks but it'll be a more relaxing hunt.
Also, assuming you're going to be hunting the eastern part of the state, you won't really be in a trophy unit and so you aren't going to need a week to hunt. You could easily tag out in 2 hours if you aren't picky on bucks. If you want to spend a full week hunting, get some doe tags, deer tags or apply for Type 6 cow elk tags.