My sympathies on suffering Lyme's Disease. I had it years ago and completely understand what you are going through, although it sounds as if your case is worse than was mine. The Dominican Republic will be great for you, with the sunshine and the wonderful people with sincere smiles.
There are ticks here, and there are even very small ticks similar to the ticks that carry Lyme's Disease. I live on the south coast. I use chickens and turkeys to keep the ticks to a minimum. To date, I have never heard of a case of Lyme's Disease in the country.
As the disease is not well known here in the country, you might have some difficulty finding a doctor who can treat you. How much Spanish do you speak? Can you adequately communicate your symptoms?
Might I make a suggestion: find some medical literature on Lyme's Disease to bring to your doctor here. Present the information in a way that does not upset your doctor: "I know this is not a disease that is common here, and my case has been a difficult one. I know you would want to look this up, so to make it easier for you, I have some information for you." If the doctor seems offended after that, get another doctor.
I recommend that you have a good exit plan, in the event you have a setback in treatment. Have enough money on hand or enough credit available that you can purchase a ticket to the US. I have friends in Miami, so I know I can get there, they can pick me up at the airport and take me directly to the hospital I have selected. And I have insurance in the US that is paid by my employer.
In general, you will find that medical care and insurance cost much less here than in the United States. Prescriptions, on the other hand, can get expensive. As my US insurance pays for medications, I have meds shipped to me via CPS.