Maybe the drawing success for nonresidents is different, but for residents almost all of the western side of the state where the public land is is pretty iffy to draw. There are lots of areas in the eastern side of the state [east of Casper] that don't fill up, you have 100% chance of drawing. Then you are dealing with private land, but if you make a scouting trip it's pretty easy to find some place to hunt. Most will charge a very reasonable trespass fee.
Yes there are areas where the gumbo can definately stop you if it gets wet. In extreme cases it builds up on your tires until the wheels won't turn anymore. Best advice is if the weather could turn wet, camp near a road that at least has a gravel surface. As far as hunting in wet weather, you can stay on improved roads, and walk from them. Antelope are plentiful enough that even walking you won't have any trouble filling your tag in a day or two. I would just leave the 4wheelers at home and bring a wheeled game carrier instead. The antelopes aren't big and the country is open so a game carrier goes real easy, and you avoid having to drag a trailer all that way, you can fit everything you need to camp in a PU or SUV. If the ground is dry it is common practice here to drive your vehicle to the game to retrieve it. Probably could get you a ticket if you did in front of a lawman but everybody does it.
Camping on BLM is legal and common. I've never had anyone bother my camp. You can't camp on State lands, the blue colored lands on your maps, but you can hunt on them.
Weather can range from glorious to kind of miserable. You could get rain or snow, but the snow won't last. Temps maybe as low as 20s at night to as high as 70s in the day.
It is usually not crowded. Like James said, avoid the first few days of the season and weekends, but don't wait too long, you will be more likely to run into bad weather.
I would suggest either butchering your game yourself or having it butchered and frozen and then hauling it home in coolers. Antelope is good eating if you take care of it. If you do the job yourself, keep the hair off your meat.