When the box and drawers have been finished by sanding through the grits, I usually go down to 400, then I usually put 2-3 coats of sanding sealer, with 400 sanding lightly between coats. After the last coat, a gentle 600 grit sand and then 2 coats of hard wax polish. Oiling is different and I am using Tru-Oil at present, which although more expensive, gives a finish with more shine. Usually 2-3 coats of oil again lightly sanding between, but I find that wiping off with a lint free cloth is better. You really need to find your own method that you are comfortable with.
You have started with a more difficult box with 6 smaller drawers and they need to be well fitting. Using the 5mm felt pads on the drawer bases helps and larger felt pads on the base of the box. Initially starting with 2-3 drawer boxes is easier and work up to the ones with smaller and multiple drawers later.
I'm looking forward to seeing more photos from you later on, but looking good so far and little mistakes are soon overcome..
Malcolm