We have that same model chest. It belonged to my mother and my daughter has it in her room now.
From where you are what I would do is first clean it with a wax and grease remover. Do it several times frequently changing rags. The wax and grease remover will soften or liquefy the furniture polish build up on the finish and the rags will absorb it.
Then get a paint sprayer if you don't already have one and mix some lacquer the sheen you want pretty thin and spray a small spot on one corner and see what happens. If it sprays smooth and level then you can scuff sand the whole chest with 220 grit or finer paper and spray the entire thing. If the finish wrinkles up like water on a freshly waxed car then stop. Google fisheye finish and see if that is what you have. If so go to a auto parts store that sells automotive paint and ask to order a product called smoothie. It's an additive you can add to the lacquer to make the finish flow out. Oddly enough the problem is caused by silicone contained in aerosol furniture polish and the smoothie is made out of silicone. The cure is the cause. Since smoothie is silicone be careful not to spray any new wood with it and be sure to dispose of any sandpaper used for between the coats sanding. You can transfer silicone from one project to another from the sandpaper or rags you might use so keep all these products separate.
Anyway when you put a fresh coat of finish over the top I think you would be amazed how much the scratches disappear. If you have some deeper scratches that are still there you can brush some lacquer on the scratches with a small narrow artist brush to fill the scratches. Then when you are working the finish you can sand the finish down that you filled with a brush and when sprayed with lacquer they will go away.