I have a newer machine that has a large tub (without the middle agitator) I took off the covers (PAIN!) and washed all but the largest one. The large ones I had to use the tub. The covers washed well! They went from stinky to spring fresh!
My neighbor has one of those. It is one of those high end computerized machines. I used it a few times while I was getting parts to repair my basic, simple, non-computerized machine. I could really put a large load her machine.
You are really jumping in there and "gettin er dun" with all of the steam cleaning and cushion rehab you've completed. Was it stinky and messy from pets or was the PO just kinda of a less than meticulous camper? We've had ours for 9 years and I've traveled with my young son and two short legged dogs...........we never had a stinky or real dirty Casita inside.
When you go down the path on getting the outside back to shinny luster and not looking chalky.......make a point of searching this forum and the other forums (www.casitaforum.com) or ( how to use either the earlier mentioned Polyglo (more expensive) or Zep floor finish #3 (wet look) this use to be called Red Max Pro (this us useful for searching process) Zep products are available at Home Depot or Lowe's. The prep prior to applying either product is the same and paying attention to the details will make a huge difference in your success or failure. It can make your Casita look brand new. In view of your efforts and zeal thus far you will be successful at working on the outside.
FYI several of us had tried using Marine wax or a good car wax after having the Casita buffed out. I had it done professionally by a local Marina guy. It lasted about 6 months and didn't even come close to looking as good as it does with Zep's product.
No product is going to last forever. Keeping it out of the sun is a big factor or keeping the outside finish. Semi-annual or annual additional coats applied will keep it looking good. The first time I applied it lasted about 3 years before I had to strip some off on the top and upper side that was in full sun all of the time.