"Rust Never Sleeps"....and a covered parking space is not always available so tarps are a very good idea! Looking in the corners at the roof/rear panel area inside for stains and around the roof vents can be excellent ways to tell if they are leaking and how bad. Then you are left with the choice of replacing rotted wood or, as many do, stopping further water intrusion and leaving well enough alone!
Checking the side panels below each side window for damaged wood is needed also...and there are two reasons the wood may show damages....one is simply due to condensing moisture on the inside of the glass and the aluminum from dripping down and secondly, once the inner and outer Pirelli seals on each side are reduced to shreds, water from the road and rain when driving can get up and under the side wall and wet that plywood. I haven't heard anything about it, but I suppose moisture could condense on the water tank and rot the floor as well so at the end of the season, drain it and try to get ALL the water out....you'll be doing a water/bleach cleaning in the spring of the tank and the lines Into it, to the faucet and to the drain tube. Maybe put a plug in the drain tube for the tank and the icebox and the sink as they may be collecting bugs and algae.
Your idea of battling moisture inside can be achieved by a dehumidifier as you mentioned or by using a 100-watt lightbulb in there burning 24/7 to warm it up. These only work AFTER you have good Pirelli seals in place so air exchange is almost non-
existent.
While you are at it, check the roof vent seals. If they are dried out and do not seal or are missing, replace them. Check and prevent it from becoming an Ant Farm or other haven for pests over the winter.