How would I say Dominican Republic in Chinese?
First of all, let's confirm the translation. If you look for "Dominican Republic" on Wikipedia, and then choose the chinese page, the result is:


  多米尼加共和国
  Duō mǐ ní jiā gòng hé guó


So, yes, it's the right translation. If you don't like Wikipedia as the only source, this is confirmed by the YouDao and CEDICT dictionaries.

I tried searching for "多米尼加" only (in both of those named above) and the result is simply "Dominica". Note that "Dominican Republic" and "Dominica" refer to two different countries, so since I think the same ambiguity is present in Chinese, you should keep the whole expression in order to be understood unambiguously.