Refugee are allowed in a country because they are persecuted/unsafe in their
HOME country, hence seek refuge from persecution/war. It is traveling to their home country which wouldn't make sense.
From USCIS :
A person with refugee or asylum status who wishes to travel
outside the United States needs a Refugee Travel Document in order
to return to the United States.In most cases, a refugee or asylee may use the Refugee Travel
Document for travel in place of a passport. The Refugee Travel Document is similar in appearance to a U.S. passport.
You
should double check with the DR embassy and an US immigration lawyer, but given that the travel document is delivered by the US and is used in place of passport, it is likely that you can travel to DR.
https://www.uscis.gov/sites/default/files/USCIS/Resources/D4en.pdf
Good luck and have a nice trip (hopefully!)