Here is the definitive answer

I am a fork,                           
I am a westerner (usually a white skinned person)

looking for chopsticks
looking for an Asian especially those countries influenced by Chinese culture.

who want to get in my drawers.
"Drawers" is a pun.  It means a boxlike compartment in furniture that can be pulled out and pushed in and it also means underpants.

This westerner is looking for some East Asian whose introduction is extremely salacious and at the same time very refined and high class way of saying.  Not crude at all.  Shows me that this person is highly educated!

I like to spoon,
Again, double meaning here.  A utensil and the slang meaning of "cuddle".  How romantic!

your place setting or mine.
Double meaning here.  A place setting in the context of spoons, forks means the way a table is set up for formal dining.  "Setting" can also mean "environment".  Here is means you can come to my place or I can go to your place.

The right dish might even score my napkin ring.
The right dish here means the right person.  Score means to "win over".  Napkin ring....  Many ways to interpret.  However, to me, it is a ring in which a male uses to prolong the pleasure.  I don't need to be crude and spell it out here explicitly.  Write me a private email if you still don't understand!  Obviously, this is a man making the gestures.

Reservations accepted! Give me a call, usually a telephone call in the US to make a reservation for dinners.

Now tell me what do you think?  Is this person highly educated or not!  Believe me, not many native speakers can be this suave and eloquent including myself!  This person is a true master of a highly skilled swordsman, wielding the English language to slash and stab to a deadly accuracy.  We all should strive to attain to his level.