You know, I wouldn't be so sure this isn't mold without more information. Black mold spores will usually wipe off a non-porous surface fairly easily, although the molds are thoroughly permeating the material by that time. K Corbin has a point with the oak + iron stain, but that stain won't wipe off with a finger; it's has to be worked at with the proper chemical.
It behooves you to accurately identify what this is because most molds are very toxic. Some researchers say mold weakens the immune system so you become susceptible to all kinds of health issues, even interfering with brain function so you cease to protect yourself appropriately. I have been there myself...
This website has a mold risk calculator that will help you get a feel for what conditions mold likes: Dew Point Calculator
Not everybody can accurately smell mold, although it typically does have a noticeable smell. Because of olfactory fatigue, even people who do notice mold will lose the scent very shortly after entering a moldy room.
If you have access to a microscope, you can scrape off some black stuff and put it on a slide. Compare to pictures available online.
By way of prevention, fix the leak(s). Then keep at least a 30-pint compressor dehumidifier running inside continuously to dry things out and keep them dry. Get one with an exterior hose connection so you can drain the bucket out automatically through a hole you drill in the floor. Or you can set it over a sink, but then the drain water goes into the holding system. Set it at 40-50% RH because the controls and gauges are usually not accurate. One class of dehumidifier only works above 64 degrees F, the other class will work down into the low 40s: the cold temperature ones are far more useful in most climates. Any dehumidifier is a 2 COP heater, so it will help keep the trailer warm also, but only when the ambient air temperature is near or above 50 degrees. (Very cold air doesn't hold much water, so the dehumidifier won't run much when it's colder than ~42 F. But mold doesn't like cold either.)
At some point you need to do some deconstruction to see how far the black stains have traveled behind things. It all has to be uncovered and dried out a a minimum.
Monitor the indoor air temp and RH with one of these: This is great hygrostat with min/max memory: https://www.ebay.com/itm/ThermoPro-D...72.m2749.l2649
Dry this trailer out, keep it dry, and educate yourselves about mold before proceeding much farther. Any object that is porous can never be completely cleaned of mold, and mold will easily spread from infected objects to anything else. But the fact is mold organisms are everywhere, just waiting the proper conditions to thrive and multiply. A compressor dehumidifier is your best friend.
There may be some scope for mold control with a quality ozone generator
, but you don't want to breathe ozone either, so it takes a good procedure and thorough airing out.