No problem about hauling/towing with that rig -
provided you don't exceed either the GVWR or GCWR of the rig.
Your GCWR is 20,000 pounds. Your GVWR is on the Federal Certification Label, probably on the driver's doorframe somewhere. That's the label that includes VIN, tire size and inflation PSI, paint and trim codes and other codes, gross axle weight ratings (GAWR), gross vehicle weight rating (GVWR), etc. It also includes the month/year the cutaway chassis was assembled.
So after you finish building your homemade motorhome, load it up with the normal stuff you'll have on the road, including clothes, bedding, food, cooler full of cool, pots and pans and dishes, full propane tanks, normal amount of fresh water you'll have when on the road, passenger(s), pet(s), etc. Then go to a truckstop that has a CAT scale, fill up with diesel, and weigh the rig.
Subtract the weight of the wet and loaded rig from the 20,000 pounds GCWR and the answer will be the max weight of any trailer you can tow without overloading your rig over the GCWR.
Subtract the weight of the wet and loaded rig from the GVWR and the answer will be the max hitch weight you can have without exceeding the GVWR of your rig. Properly-loaded tag trailers have 10 to 12 percent hitch weight, so if your loaded trailer grosses 8,000 pounds, figure on 800 to 950 pounds hitch weight.
The only motorhome receivers I see are rated a max of 5,000 pounds weight carrying and 6,000 pounds with a weight-distributing hitch. But you don't want to be caught dead without a weight-distributing hitch, so plan on a max weight of car and trailer of 6,000 pounds.
Motor Home/RV Receiver Hitch
You have a 4R100 tranny with 4 gears plus a lockup torque converter. One of those "shifts" you counted was the torque converter locking up.
Your 4R100 tranny is designed to tow in overdrive. If it can't make the grade without downshifting, it will downshift. As a general rule, the only time you want to lock out overdrive is when coming down the mountain with a heavy load pushing on you, or in up-and-down hills where the tranny shifts often. It doesn't hurt the tranny to shift up and down, but it causes extra heat that could overheat the tranny.
I would not tow more than a rowboat without a tranny temp gauge. Any tranny temp less than 200° is normal. 200° to 210° is okay. 210° to 225° means you are working the tranny very hard and probably at a slower speed so the torque converter is unlocked and not much air is passing through the tranny coolers. 225° is the red line. Don't go there.