Not very many private CG's on the coast. Your biggest hurdle will be your 40' MH. There are many St CG's along the way but some are not big enough for a 40' MH. But the majority are. But not all sites in the majority will work for a 40' MH. In May during the week you'll have no problem getting a spot somewhere. It's the weekends that can be tricky. Some of the St parks have walkup only sites along with reserved sites. Those would be the ones to grab on a Thur or Fri early. The Oregon coast is very popular during the summer but that's starts around Memorial Day weekend. So anytime before Memorial Day is better. We live in Oregon and don't even think about going to the coast after Memorial day thru Labor Day. It's a zoo.
There are also some nice RV parks along the way that can be utilized if no St park spots are available.
Sept is different than May. Sept is one of the best times to go to the Oregon coast. Weather is some of the nicest that month, so yes it will be fuller in Sept. May weather can is a******shoot.