First, annexation lost be more than one vote because a treaty would require 2/3 to pass.
Annexation does not automatically mean statehood. It might remain as a territory for many years, and Congress would need to approve of any new states. If annexed, Santo Domingo might remain a territory, become something similar to a commonwealth like Puerto Rico is, be given its independence, or become a state after the Civil Rights era.
The first stage would be American business investment. Roads, ports, and other infrastructure would be built, as well as American business plantations for sugar cane, tobacco, and coffee. Likely, much of the business investment that later when to Cuba would go to the Dominican Republic. Eventually some light manufacturing will develop based on local plantation needs and mining. During this time, there is going to be a lot of poltiical tension from the existing Dominican elites and any new American bureaucrats. I do not see Santo Domingo becoming a state at this time.
I don't think the US possession of Santo Domingo would change the Spanish American War or its results. Cubans were never in favor of US annexation, and I think direct possession would be opposed even within the US.
I don't see many freed blacks going there. There simply isn't the industrial jobs that attracted blacks to northern states later on. Nor do I think is there enough available land for independent homesteaders. Instead, we may see the "black intellectual" class - Booker T Washington, W E B Du Bois, Marcus Garvey and the like - make the area part of the black cultural renaissance which was based in Harlem IOTL. There would probably be a greater identification with Afro-Caribbean culture with Black American culture. One possible side effect would be to push the more Latin Santo Domingans into identifying as "white". Visting white Americans from the states may start including them as being "white" too (or at least some of them), which would likely poison native race relations.
The second stage would begin around World War I as Santo Domingans would be recruited for the war. We would probably see the beginning of the US giving Santo Domingo full control over its internal matters. It might be upgraded from a Territory to some kind of commonwealth status, but I don't see them becoming a state. We would probably see the development of local universities and other social institutions. The elites would be heavily Americanized and English speaking. Everyone else would probably be taught English but still use Spanish as the everyday language. If Haiti is occupied, then the occupation forces will likely be commanded from a US base in Santo Domingo. Likewise, any military operations in the Caribbean will probably be based out of there.
Culturally, the inclusion of the Santo Domingan Catholic Church with American diocese might be a major factor in integrating Dominican identity with American identity. You would see Dominican priests and bishops work with their Irish, Italian, and Polish brethren in the US which would provide a unwitting source of integration.
The third stage would begin around World War II. There would likely be large investment in air and naval bases to secure the Caribbean against German U-Boats. Greater mobilization of Santo Domingans in the armed forces would create a strong bond between them and their fellow citizens. The economy would continue to develop. In the postwar period, the city of Santo Domingo may become the entreport of the Caribbean Sea, replacing Miami IOTL. Tourism booms.
At this point, I see either a drive for Santo Domingo to become a state or become independent. The drive for statehood would likely be linked to the rise of the Civil Rights Movement if it is not granted in the 1950s at the same time as Hawaii and Alaska. if it is delayed, it might become a state by the 1970s, or become so disgusted that it seeks independence.
The problem for Dominican statehood is the same problem with Puerto Rican statehood - the people likely do not speak English as their primary language. That would be a big barrier.
I'm sure many specifics can be argued, but this is the general development I see if it was annexed by the US back in 1871. Much depends on what kind of specifics would happen.