Well, if you have never been here, you may be in for a shock. What with the garbage and the hookers and the stray dogs. As to the North Coast... I do not know most of the towns but it is far away from the Capital, far away from cultural activities, and will not perhaps have that much company for your Mom. Also it does not have the best medical care, which is located either in the Capital or Santiago.
Here in the Capital, for instance, there are regular activities from the Embassy of Spain and many Spaniards, as well as the International Womne's Club, and the English library, among other things. There she could perhaps make friends with others of her age group and interests.
I am not saying that she might not adapt to life in Puerto Plata or Sosua or Las Terrenas, which has a large expat community from many countries, but they are primarily beach resort towns and after the beach, there is not that much to do. I think that there is a movie theater now in Puerto Plata, but she will not find theater, or malls or much else. Plus, she will be car dependent.. unless she is good at riding on the back of a concho.
There are parts of Juan Dolio that are nice// the little village is charming. But most of the apartment buildings are empty and there is a sense of a ghost town about it.
Bavaro might suit her, and you, but again, there is the car issue. Evidently there is a growing community outside the resorts of both educated Dominicans of middle management and foreigners. That might be your best bet in terms of both job opportunities and being close to the beach. There is also a branch of one of the biggest hospitals there. It is flat land, beaches for miles and miles, and not a lot going on, but it is near some fairly big towns.
Santo Domingo is right on the water, and although there are a couple of beaches, I would not swim in the water there. The nearest really good beach, Guayacanes, is forty minutes by car.. and car is really the best way to get there.
Rents really depend on where you live. Three bedrooms are harder to find than the two bedrooms as family size has gotten smaller. Go to supercasas.com and take a look. If you want to stay under the one third for rent, you probably can find things. And I know that everyone here says rent rent rent.. which you certainly need to do for the first year. But they are pretty strict about the rent increases in the Capital which go up either five or ten percent a year. I would suggest that you start out looking for a two bedroom, just to get your feet on the ground. Also, you would be able to buy a two bedroom apartment within your 100k price range.. an older one, perhaps, but still in a good neighborhood.
I am an elder myself and lived on the beach for two years or so and found that I was sort of shriveling up for lack of intellectual companionship. A single woman is not invited out that much, except, of course, by the much younger Dominican men who are looking for a Sugar Mama. Plus Dominicans are very family oriented and tend to avoid tourists in general and newcomers. It will take a while to make any friends and ex pat gathering clubs are good places.
But the most important thing is for you to find work, as you need the money and be bored out of your skull if you do not work. You need to start the residency process back in States... Google DR1 residency and you will find our 18,000 threads on that subject. I suspect that hotels will be your best bet, but the salaries will depress you. Do not expect more than US $500 a month and hope to work your way up. If you have experience, or classes or whatever, you will have a better chance but they do really need English speakers. It will not give you a living to start but you are in a situation which might allow you to take something less and grow.
The dream of a villa with three bedrooms by the beach in your price range, is, I think, long gone. Perhaps in a more isolated place in the Island, like Pedernales or Luperon, you might find something like that but it would be hard for you to find work.
Those are just my thoughts. First you have to come see the place. It is an Afro Latin Carribean island, the Spanish spoken here is not the Spanish of Spain, and many Europeans (I am just saying this in general, not about your Mom) seem to think that the place is "beneath them". (I am trying to find a nice way to put that. Spaniards are not particularly personally beloved here even though they are worshipped in principal. One of my friends said "in they end, they always remind you that they were the colonial rulers".
So come and see the place. But also consider perhaps other countries where the living is even cheaper.... $2800 is good here but it is nothing to write home about.. it will not bring a Huge lifestyle.. a good one, certainly, for one person. Local health insurance is primarily available only if one buys it before the age of 65, although there are companies who will write policies for other elders if they are in good health.
I do not want to discourage you. I came here 10 years ago and it was the best move of my life. But I knew the Caribbean well and had also looked and travelled in Central America and eliminated several countries before I came.
If you are only coming for two weeks, I would suggest that you stay in the Capital for at least three or four days, then rent a car and drive the North Coast a bit, from Puerto Plata to Las Terrenas, and see how you like it. Then, next trip, you can perhaps stay at an A1 in Punta Cana. Like that.. paso a paso