I'm actually doing a similar repair right now. Question: have you removed the plywood floor/bed yet? Mine is glued to the exterior plastic. It's not coming out without a MAJOR overhaul and its not so bad it can't be remedied. I plan to remove the rot/soft material, treat with preservative, treat with wood hardener then skin over with 1/2" CDX.
Liquid nails/ subfloor adhesive had bridging and flexing qualities. I've built homes for 40 years and IMHO it is the proper adhesive for this application. Titebond is a great glue- for furniture. It won't flex at all and requires clamping that is impractical for what you're trying to do. I also plan to install the new plywood at a 90 degree to the existing. This will deal with any joint issues. I think you will find, as Da Hose did, that there is a joint there somewhere. I can't imagine getting such a large piece as you mention in the door and installed.
Regarding your brakes- proportion valves work front to back, not side to side. It's either an adjustment or possibly as you suspect a faulty cylinder. Try adjusting again. I check mine with an infrared heat sensor. Cheap, $15.