Brother lable makers are pretty cool. It's a clear or colored rectangle/square sticker without some work though. I don't recall what the smallest font or tape size is but they are kinda nice. Smooth, shines; more like painted steel.
I think a high gloss paper is what I'd try using. Your possibly seeing paper grain and that could be part of the real issue; it should not have graining. It should be smooth and flat as can be.
On some thinner papers your background color could possibly effect the overall tone light or dark or a color tint. So if you're getting a washed out look to overlay lettering, you either have to darken the whole base background, or do some loose fill color on the base surface , by hand to help darken it.
For a surfaced wood edge I’d laminate and then trim at a beveled angle with new exacto blade and straight edge; so the face is ever so slightly smaller than the back, then use brown edges. Metal I'd cut for90° edges and use black and or silver edges. It would be hard to align the precut paper and get nice flush edges and no "step" at the papers edge. I consider a burnishing roller my buddy for laminating; from graphics to countertops