Hi Sorin,
The fine lines will be from each layer you apply of the melamine., they will get worse with each layer.
I rarely sand above 400 grit to start with as that gives the surface melamine a good key layer. (Depends on the wood)
I moved to melamine a few years ago no and many complain about finishing and removing the layers, that’s mainly because you haven’t given the melamine enough time to harden. It needs at least 24 hours before you can use a grit type polish to take out the lines. (Or the UV method above)
There are as many ways to finish a pen as there are turners.
My options (it’s only the way I finish my pens)
Acrylics are sand up to 1200 with the lathe running and horizontally with the lathe stopped on every grade. Final grade is horizontally only.
Then use Yorkshire grit, microfine and then a final polish with any car polish you fancy.
There is a little trick you can do to speed up the hardening of the lacquer ! I use a UV nail dryer that has the four UV lights in and place it above the pen for a few minutes (rotate the lathe by hand while it’s on) each layer added, then I sand very lightly lengthways with the lathe stopped with a scotch fine cloth to take out most of the lines.
As above, I always finish with YG fine, (which looks like I’m choosing something else when this tub runs out) to take out the lines and polish to a shine.
Woods and other organics.
Finish to 400/600 (depending on the material) then melamine lacquer using a slightly adapted Phil Dart method. For a high gloss leave overnight or 24hours to allow the lacquer to harden. Then Yorkshire grit microfine to bring the finish to a high gloss.
Wood/acrylic mix (so Called Hybrid blanks)
Sand to 600, lathe stopped and horizontally on all grades. Then use melamine lacquer using the slightly adapted Phil Dart method. Tend not to use Yorkshire grit on these as I have found that occasionally I have had residue get between where the wood meets acrylic and leave a fine white line. Which is a bugger to remedy.