Most trailers' molding is just stapled in place by the manufacturer.
I prefer to drill and use paneling nails like one would in a home. they hold better, in my opinion.
I also ran a bead of silicone around the bottom of cabinets to keep spilled water from intruding under the cabinets in the Galley, but let it cure before putting on the molding, in case I ever want to remove the molding easily again. then fill the holes with wood putty in matching color.
My friend has a Sunnybrook TT of about the same vintage as yours and is fixing up her interior too. Helped her put reflective UV coating on the inside of the windows so I really got to crawl around in it. Very nice trailer, very well laid out.
2011 Heartland Road 361 Warrior Toy Hauler
Ford F350 4X4 Tow/Haul Tonner Dually