Why do these lines only show up after I finish the wood?

Long, fairly straight lines are often left by surfacing or planing, rather than caused by uneven pressure from the sander. Finish makes shallow marks easier to see. Before finishing, wipe a small area with water or mineral spirits and look across it in raking light. That gives you a preview of what the finish may highlight.

If the marks are actual grooves, a scraper or hand plane can remove them, or you can sand them out starting with a grit coarse enough to flatten the surface. Then work up through the grits, making sure each pass removes the scratches from the previous one. Pencil marks are a handy guide for seeing whether you’ve covered the whole area.

I wouldn’t jump straight to finer paper. If the lines are deep enough to catch the light, finer grits can just polish around them. First get the surface flat with a scraper, plane, or a flat sanding block and coarser paper. A random-orbit sander is useful for the later smoothing, but it can be slow work if it has to remove noticeable surfacing marks.

One thing to check is whether every line is really a defect. Figure in wood can shift in appearance as the light moves, but repeated straight lines that stay in the same place are more suggestive of tool or sanding marks. Look from a few angles before removing more material.