Check out Kohler's Caxton #2209. The outside dimensions are 17x15. The inside bowl dimensions are 15x12. I have this sink in a 21" deep vanity. I have a granite backsplash that is 1-1/2 deep. From the backsplash to the sink edge I have 6 inches of space for the faucet. If you use a tile backsplash it will not be as thick as a my stone backsplash and you should have at least 4" of space with your 18 vanity, for your faucet. Remember that the top for your vanity will probably be about 19 to 19-1/2 inches deep.

Maybe this picture will help you visualize how the Caxton #2209 could work.

If you get a single hole faucet you do not need the escutcheon plate you see in my picture. We had to use it because the plumber mistakenly told the fabricator that we were using a 4" center set faucet and they drilled 3 holes. The escutcheon plate covers the extra holes. I think single hole faucet look better without that plate.

Two suggestions:

1) Get an oval sink in a surface basin style.

2) Get a single handle faucet and mount it in back, but off to the side.

This worked for me with just about the same depth vanity.

Have you considered a wall-mounted faucet? That would eliminate the need for space behind the sink and any sink would work with your 18" vanity.

I'm going the wall-mounted route but it fits my design style. And it's easier to clean, which can be tough in tight spaces.