Can you help save re-staining project please
The purpose of the solvent is to not remove but redistribute the stain that is already applied, although it will remove some of it. It will not remove what has penetrated into the grain of the wood, but it will lighten it. The solvent may even push it deeper into the grain, and that is ok too.

You can sand your project at this point, the scratches will still most likely be holding stain so you can see how far you need to go on your sanding. If you remove more stain by sanding, that should not be a problem. Presumably the lacquer thinner has evaporated before you sand, and the rest of the areas have been cleared of "excess" stain, that is, stain that has built up excessively on the surface, and all the surface has, for the most part uniform coloration throughout, before you sand.

As I mentioned in my comment earlier you can sand before the lacquer thinner, but your sandpaper will readily clog. Then you could use solvent to re-disperse the stain over the freshly sanded surface too and let it dry, Then apply the second coat of stain, rubbing it out after a few minutes, again, a VERY important step.

Which ever way you choose, the idea is to get rid of the cross scratches, disperse the heavy deposits of stain using solvent, let it dry thoroughly, and recoat with fresh stain, and rub it out after a few minutes. Rubbing out the stain removes the heavy pigments off the surface, so the clear coat you choose to use will adhere. If you do not rub it out, what you lay on as a finish will peel off in time. probably in sheets.

One last thing, if the stain still looks lighter in places than others, you can still add another coat of stain, the places that are dark will not take as much, and the lighter places will take in more stain, essentially evening out the color. After all, wood can only take on only so much pigment. You can even try localizing your added stain, only adding it to the panel or just a rail or a stile, following the joinery in the doors.