2-3 months in the DR

Cars like a Kia Picanto get good gas mileage. They are common rentals here.

I would pick a few destinations so you can see various locations without a lot of round trip travelling. Caribe Tours and Metro Bus are very practical ways to get to different areas.

As for your monthly expenses as a single person that depends greatly on so many factors during a short stay it is hard to estimate.

I also sent you a private message.

I don't think you can get 30-40 MPG in any car here in DR. Fuel is of poor quality, so cars that are supposed to give you that mileage per gallon usually end up giving you 20-22 at most.

then, get a car which works with propane....

There aren't many places you can't get to (or really close to) using Caribe or Metro, as well as the smaller regional buses. Unless you want to stop in every town, or do some serious off-the-beaten-path exploring, those are the best and cheapest ways of getting around. In addition, local cab service is very inexpensive.

In terms of seeing the country, it depends on what your interest is. The usual suspect "touristy" spots include

Santo Domingo and Juan Dolio to the south;

La Romana and Punta Cana to the east;

Puerto Plata, Sosua, Cabarete, Rio San Juan, Cabrera and Samana to the North(east); and

Jarabacoa and Santiago in the middle.

All are nice, fun, generally touristy spots.

I've grouped them together because, with the exception of Samana, they can all be visited with short trips from a central city. For example, if you stay in Cabarete, you can easily do the rest of the north coast towns, none of which are more than an hour away.

Most all towns have "apartahotels" which are basically extra large hotel rooms with kitchenettes. You'll find they are cheaper than hotel rooms if you stay for more than a few days. This will likely be less expensive than renting a car and paying full price for an ever-changing list of hotels.

However, if that's what you want, we can probably help there, too.

Be very careful and think twice (and then think twice again!) about a car. I believe your USA (Alaska, right?) drivers license is only valid here in the DR for the first 30 days of your visit (and it sounds like you've already exceeded that time period).
Should you be stopped by the police, or even worse, be involved in an accident, the lack of a valid drivers license could be a big problem for you.
Besides, if you want to live frugally, travel like the locals do. Bus, gua-gua, and caritto. Cheap and fairly efficient (and the Metro and Caribe Tout busses are even comfortable!).
Enjoy your travels.