It seems from here you might be taking to big of a first bite. A quality guided bull elk hunt for 2 will put a big dent in $10,000.
There are some guided cow elk hunts
, that might work.
Antelope in my mind make a better first non-resident hunt. Guided can open up some private land areas, that greatly improve your odds.
If it's an area that multiple doe tags can be had, you'll learn a great deal about LRH in a short period of time.