How to paint my old oak vanity to look like a dark wood??

Take a look at these two paints:

https://www.google.com/aclk?sa=l&ai=DChcSEwjruuCiverdAhUDbn4KHR_VBOEYABAKGgJwYw&sig=AOD64_0-oSLpmF7jfPTjpMBr2YAHrJURKg&ctype=5&q=&ved=0ahUKEwiXuNeiverdAhUNCKwKHa1eDt0Qww8IMA&adurl=

https://www.google.com/aclk?sa=l&ai=DChcSEwjruuCiverdAhUDbn4KHR_VBOEYABAKGgJwYw&sig=AOD64_0-oSLpmF7jfPTjpMBr2YAHrJURKg&ctype=5&q=&ved=0ahUKEwiXuNeiverdAhUNCKwKHa1eDt0Qww8IMA&adurl=

Take a photo to the paint department of your home improvement store and ask for their advice. They have the newest materials and application tools for a professional project.

Sand it lightly and apply gel stain.

Rustoleum makes a kit for bath vanities. I used their kit for my kitchen cabinets and it's great. You can see what I did here: https://www.flippingtheflip.com/2015/09/kitchen-cabinets-black.html

My fave is the gel stain type paint

You can stain over stain with a good gel stain. Any brand will do. You can go darker but not lighter.

https://www.minwax.com/how-to-finish-wood/change-stain-color-with-polyshades/

Its easy to make your piece look like darker wood with Dixie Belle "No Pain Gel Stain" . You just clean your cabinets really well with Dixie Belle White Lightning. Then with a clean cloth apply a very light coat of stain. Work one direction and do not go back and forth. We have some great tutorial SHORT videos in our Dixie Belle Facebook group.

I am attaching some kitchen cabinets (before and after) that were stained with Dixie Belle No Pain Gel Stain in Espresso by our retailer, Deborah Hubbard Fiene . Hope this helps

https://www.facebook.com/groups/FriendsWhoLikeDixieBellePaint/