Will, like you we really prefer Forest Service sites. If you get a unit that is over 30' there definitely will be established FS campgrounds you can't get in. Then again, there are campgrounds you can't use if you are 24'! That doesn't mean you can't find a campground you can fit in since bigger sites definitely exist. We live in Utah and travel mostly in Utah, Idaho, Wyoming, and Montana. We have a 30' travel trailer (tongue to bumper). We find about half the FS campgrounds in those areas won't hold us. In fact there are a number that are tents only! But half of the campgrounds do hold us, and most of the ones that hold our 30' would hold a 32 or 34'. For boondocking on FS or BLM land outside of established campgrounds, you will have no problem finding a spot no matter what your size.
The biggest problem we have is in NPS campgrounds. Big sites do exist but they are fewer in number. Many of the NPS campgrounds were built by the CCC in the 30s! Get reservations early or show up very early in the day to get a first come/first serve campsite if you are big and travel during prime season. We have a tendency to go to the popular places early or late in the season then head to the forests and mountains away from the crowd during the summer vacation time. Also remember that there are private RV parks outside all the national parks that are very convenient and still let you see the beautiful places.
If you want to dry camp or boondock, get a unit with the biggest tanks you can - fresh, gray, and black. We have 80/80/40 and the two of us can easily go without dumping for a week while still taking short showers daily and washing dishes. Also you will need a lot of solar or a generator plus good batteries. I love heading into the forest and relaxing for a full week, taking time to sightsee, fish, walk, and just enjoy the area.