We have taken two three month long trips a year since 2007. We go from the East Coast South in the Spring and West in the Fall. We usually have the farthest point in mind when we start out and pick some potential points of interest along the way. We generally travel 200-250 miles per day (10AM-2PM). We prefer to stay off the interstates except to get through/around the larger cities.
In the evening, I look at the route for the next day and find some potential stops. About noon, we call to check on availability. At check-in, we ask what there is to do in the local area. On a few occasions, an overnight has turned into a several day stay featuring visits to local attractions. We try to avoid travelling on weekends if possible and may make a reservation for Friday/Saturday in advance. In rural areas, city/town parks may allow overnights and some even have sites with hookups. We have had many great experiences in these public areas. Local events like home games in college towns, county fairs, etc. can have an impact on the local availability of campsites.
In my experience, finding overnight parks has become more difficult in the past couple of years. Last year finding overnight stops was compounded by the Covid travel restricts such as closures, non-resident bans, quarantine requirements, etc. Our last trailer came with a Thousand Trails Zone Pass. Using those parks often requires making a reservation as soon as the 60 day window opens.