They’ll need an unusual postal abbreviation; South Dakota took the obvious one.
Would it even be called the State of Santo Domingo? Quisqueya might be more fitting given how having a Santo Domingo City and State of Santo Domingo could be confusing (pay no mind to NYC, Jersey City, Kansas City, etc).
Otherwise, SG might work. Or ST (Short for Saint, as in Santo Domingo).
Grant insisted on Santo Domingo being admitted to the Union as a full state. You'd have a state run by brown-skinned Catholics that'd likely be loyal to the Republican Party for the first half-century of its existence. There'd be Spanish-Speaking Hispanic Catholic Republic House and Senate members in 19th and 20th Century America. Weird.
If the US has SD, and likely had to fight an insurgency to keep it considering there were still lots of armed people around after the Spanish War, we might not see the annexation of Puerto Rico. On the other hand, the Grito de Lares in PR probably would flee to SD as well as NY (where they fled OTL) and that'd make Puerto Rico a bigger issue in US politics sooner than OTL.
If the US still annexes PR, they might be admitted as a state quickly.
There'd likely be resettlement of freedmen in Santo Domingo. Grant (and Frederick Douglass, among others) was for that, as he felt that it wasn't safe for many to remain in the south.
The US would have a big naval base in Samana Bay. I can see Santo Domingo becoming a significant city. Maybe it becomes the "Capital of Latin America" instead of Miami.
I can see a lot of the same forces that caused sunbelt migration to Florida OTL (and, briefly, the Isle of Pines in Cuba) to apply to Santo Domingo TTL.
Dominican Cowboy movies might be a thing. Ranching is big in the country.