You should be able to do the visa for your passport yourself. Call the Embassy in NY and make sure you have all the required documents, all the required apostile stamps, all of the documents are valid age wise as per the criteria provided by the Embassy, all documents are translated into Spanish (or you may be able to have the Embassy do that for you for a fee).
At least for your first residency application in the DR, I would recommend a lawyer, especially if your Spanish language skills are not robust. Don't wait, get your lawyer in the DR sorted first, they can often help with valuable info for the visa process back home.
Get cracking on your FBI criminal record check. That can sometimes take a while to get back. That document should be valid for 1 year.
I believe the DR considers long form birth certificates valid for 6 months from the date of issue.
Do you need a new passport? It should to be valid for a couple more years at the time of your application in the DR. If you do get a new one, keep the old one as immigration will probably want to see it.
Confirm everything you are told several times.
Good luck.