If you speak some Spanish, this is easy to do yourself. The forms are freely available in the office in Santo Domingo. On a podium in the middle of the office, you will find forms as well as a list that tells you the requirements. So, get everything together in good order. Go in early one morning with your package of documentation, photos and other requirements all together. First, go to the cashier and pay your money. (I forget now how much it is). The cashier will tell you what to do next. Basically you wait until they've processed your documentation, then go for another medical examination. Then you go back into the main office with your medical included in your batch of documentation, line up, wait and eventually you'll get a residency, if they have plastic available to do the card. If not, you need to request a paper that tells everyone you're legal. For us, this took one day, but all of our paperwork were in order and we had everything ready.
Don't do this if you do not have sufficient Spanish to understand instructions, can fill in Spanish forms and find your way around Santo Domingo. In this case, it is better and easier to let a lawyer do it for you. Use a reputable laywer and not some fellow off the streets.