What’s the best way to DIY an upgrade to light oak bathroom cabinets

You can try gel stain, it is supposed to adhere well:

I painted mine gray. I scrubbed them first, then sanded. Once the finish was dulled, I primed and painted them. I had intended it as a short time fix, but, almost five years later, I still like them.

You can easily stain them. Remove from wall if you can. Clean & sand off any finish, wax, dirt etc. You can buy stain or use shoe polish or make a stain from steel wool (with no soap) rusting in a small glass container of water. Apply and wipe off stain until you are happy with the color. Letting it sit awhile before wiping off may help make it darker. When dry finish with varnish or polyurethane or oil, 3 coats minimum with light sanding between ( ultra fine grit if you want shine) Remember that if you use linseed oil the rags can self-ignite so they need to dry out safely or go into fireproof closed cans. I usually spread them out on a branch, wall, anything that will allow save airflow for drying. They can go into regular trash after they are dry.

Wipe on gel stain - great stuff and so easy to use!