Presuming your trailer is still wearing its original gelcoat, then you can try a few things to see if you can bring it back. At some point, most gelcoat gets to the point where it's better to paint
(and a good, two-part paint
job is a thing of long-lasting beauty); but on the other hand, much can be brought back, and the original gelcoat is nice to save while it is still feasible (until the point of no return, but you may not be there yet).
Black spots are probably some kind of mold or mildew, and if your gelcoat has become porous and rough, of course it tends to harbor these things.
Washing and waxing will only clean and protect what you have; it won't bring shine that doesn't currently exist.
Think of it a bit like skin. What you want to do is try to slough off some of the dead outer layers, without going through the gelcoat (only ultimately you can paint
fiberglass, which doesn't really work with skin!).
Examples of products you can use are rubbing compounds, and wetsanding paper. Rubbing/polishing compounds are like "grainy" creams that you put on with a buffer (unless you have Super Arm); wetsanding paper is just like any sanding, but you dip the (special) paper in water to lubricate it. With either of these, you should see the gelcoat getting darker and shinier (it gets darker as you remove the light-colored oxidation). Sometimes you can find an area under a sticker or something that was attached to the trailer to see what your "virgin" gelcoat looked like.
One note is that I have not had good luck with "consumer" type buffers. I found that I needed a bit more "oomph" than they could give to really get the shine back.
Once you have fresh, shiny gelcoat exposed, then a coating of a good wax will help protect it.
Some companies that make various grades of compound: Meguiars and 3M
Wetsanding paper: You might make use of grits from 400 to 1500, with higher numbers being smoother. I would start with something a bit higher (maybe 800-ish?) and then work down to a rougher grit just to see what to start with (this way you don't start *too* rough without realizing it). Then once you have done that, you can work back through the grits to the smoothest one you are happy with.
One nice thing with the wetsanding paper is that for a very small investment (just a few sheets of paper and your arm) you can experiment on a few areas to see what your chances of a good "rebirth" are.