Welcome from Arizona and Colorado. I am retired and split time between two states.
I would suggest you look into hunting pronghorn in Wyoming. Pronghorn are quite striking if were successful on a hunt and decided to shoulder mount the buck. As with many of the states that have several varieties of big game, Wyoming has a lottery system to obtain tags to hunt big game. As a non-resident, the majority of tags are not available as are reserved for residents of Wyoming. Tags open for non-residents to draw get separated into "buckets" and may to your benefit if have been unsuccessfully applying for years as will have better odds but then some buckets are the same odds for every applicant. You also pay more if you obtain a tag as a non-resident. For example, a bison tag when a resident is about $400 while you and I would pay about $6000. Yes, 15 times as much.
Some states have transferable licenses where the original holder of the license can offer to sell it to someone else. You could buy a tag and hunt that season.
Some states have tags which can be obtained without participating in a draw. You pay and have a tag. Sometimes is a cap on those tags so first-come first-served.
Some animals are in states where were not initially found so may be declared exotics and can be hunted without being in the lottery draw in that state. Texas has lots opportunity to go shoot exotics.
Most states will require you have passed a Hunter's Education course before can obtain a big game tag. Fortunately, to my knowledge, all states in America accept a Hunter's Education from any state so only need to take the class once.
One more observation. America is big. So big that if you get turned around while hiking in some of our forests that you could walk in a straight line for days and not cross a road or see anyone that could help you. Our weather, especially in September to December, can drop from t-shirt weather at midday to by 3pm you better be have layers of proper clothing to put on as the temperature falls and cold rain/wind/sleet/snow tries to soak you into hypothermia. Every year hunters die from hypothermia as well as lightning strikes while hunting in mountains and forests. Also, you are not the apex predator if enter grizzly bear habitat and shooting one in self-defense will still be a headache to resolve and could result in fines or needing to hire a lawyer. We also have a handful of venomous snakes that rarely are lethal but will ruin your day or hunt.
I spent a summer in Romsey long ago before the White Horse pub became a money-laundering front for Russians. Was wonderful and rode trains to Bath, London, Stratford and Warwick.
Spent a month riding trains last summer after the QM2 landed on our sail over from New York City. Spend a few nights each in Southhampton, Winchester, Cardiff, York, Edinburgh and GlenEagle. A train strike was underway but very polite as would announce the labor stoppage the prior day so was less chaotic as got to the train station. In America, would likely be trains on fire during a similar labor stoppage and violence if substitute workers were brought in to get the trains running. You might get assaulted if wanted to use the train as a passenger.
Our dog went with us. Picture is of Cardiff.
View attachment 409271