If you are going to travel, campground fees and fuel costs can be the two largest budget items. They are also controllable. The average cost of our 247 stays in developed campgrounds over the past year and a half was $27.10. However, we do not go to prime snowbird areas or other areas in prime season like you will be doing. Dry camping/boondocking can save on camping fees. If your RV is not already setup with a large battery bank and robust solar system, doing so for extended periods of time may require a fairly large investment up front unless you want to run a generator a lot or just not use the amenities of the RV.
Like one of the other posters, we tend to drive about the same number of miles with and without the RV when in sight seeing mode. Local or regional fuel prices can vary considerably. We have observed as much as a dollar per gallon difference within a week of travel. Some areas like parts of California are notoriously high. Fuel price websites like gas buddy can give you a reality check on prices in the areas you plan to visit.
You don't say how you arrived at your entertainment budget. We spend far less, but as seniors with the America the Beautiful Senior Pass, entrance fees/day use fees to most federal facilities are free. We also buy state park access passes when we will be in a particular state for an extended period of time. We have found that many of the smaller free or low cost local/county museums are just as good as the big fancy higher cost ones.
Everyone has different preferences in how they like to travel. To us the point of traveling is to see and experience new things. If we have to worry about every penny that we will spend or do things that we don't enjoy like bathing in the sinks at Walmart; we would rather just not go and save until we can travel the way we like to.