How do you call an exceptionally black Afrian-American as opposed to those with lighter skin? Is there such a word in American English or slang?
As has basically been said dark vs. fair would do:
(Of someone’s skin, hair, or eyes) brown or black in colour.
(Of a person) having a light complexion or hair.
These terms have comparative (darker, fairer) and superlative (darkest, fairest) forms and can be used of people of any complexion. So you could describe a white person with brown hair as being darker than a redhead.
So you could, if you had reason to, describe someone as simply a very dark African-American (though "African-American" is really not very common outside of America). Although it is worth noting that there be dragons. However, it is not necessarily politically incorrect or offensive at all and such terms can obviously be very useful if discussing racial politics or prejudice or whatever.
As a point of interest.
I grew up in London, England and I spent a lot of time in my teens (about five years ago now) with members of a community in which fairness is still often considered desirable and attractive. The slang term they used to use for dark people was "blick" - which has an urban dictionary entry, but is not very common, it seems. I do not recommend using this word unless you know what you're doing. But, as a matter of linguistic interest, I thought it worth mentioning.
We generally just use the terms:
"Dark-skinned black person/African American"
"Light[er]-skinned black person/African American"
Here are some examples in articles:
My mother, the first woman I ever knew, loved and admired, was a dark-skinned black woman and, to me, the epitome of beauty and glamour.
Type dark skin or light skin into Twitter and you will see colourism in action. One tweet: "Party on Friday. White Girls free. Light skin girls 5dollars. 50 dollars for dark skin girls".
I've always wondered why I wasn't born with the dark skin I thought I deserved to have. My mother is a beautiful shade of brown. My father, who passed away when I was a young girl, was a dark shade of chocolate. [...]
It hasn't been easy for me as a light-skinned African-American woman, and this wannabe black woman Rachel Dolezal is making it that much harder for me to live my best light-skinned black woman life.
There is not really any term that I know of that would be polite to use to someone's face, or in casual reference. In a literary context, you might refer to someone as having dark ebony, obsidian, or pitch-black skin, though in terms of actual color the last two terms are misnomers.