There are quite a few opportunities for free range exotics on public land here in Texas. Lake Amistad NRA has aoudad and muflon sheep in addition to native whitetail, hogs, and javalina. It is archery only but it is walk on with no drawing required.
One of my favorite exotic hunts here is for nilgai in deep south Texas. Laguna Atascosa NWR, and the two sites of Lower Rio Grand Valley NWR have draw hunts for nilgai. There are rifle hunts but it is much easier to get drawn for the archery hunts. Several other draw hunts on state land have aoudad, blackbuck, gemsbock, oryx, axis deer, fallow deer, sambar deer, and of course feral hog. One of the state WMAs even has warthogs running around that they would love to get rid of.