For antelope, Wyoming is probably the best bet. Not really any over the counter tags but in many units there are plenty of tags that don't ever sell out and can be drawn by a non-res no problem. Or I guess you could call them over the counter since after the draw all left-overs are sold 1st come.
The units that don't sell out usually have lots of private land but if you do your homework you can find places to hunt and should have no problem getting a antelope anyway, deer is harder.
Call the WDFW office for whatever area you want to hunt, talk to the bio and get the list many will have that gives you names of landowners/ranchers that allow trespass hunting, some are not much and some are, just have to call and find out.
Or better yet, check the WDFW website and look for the units with Hunter Walk-in areas, this is land that is private but you can access, plenty of it to be had and if you go later in the season some can be OK for getting at least a OK buck - fun hunting whatever you do and be sure to pick up a couple doe tags while your at it.