Nevertheless, I checked the law online and it actually says something different. It says the foreign license is only valid for 90 days.
What is good to know is that one can legally apply for a driver's license without having to get a residence. So one could actually get a license in less than a week -- it sends one to the Consulate of the country that issued the foreign license, then that probably has to be legalized, and then one gets to apply for the Dominican license. This is a procedure that does not seem to require a lawyer. One can do all the steps by oneself.
On the other hand, from what the person at the Casa del Conductor tells us, apparently it is a practice to allow those who keep their legal status to drive. But to be extra safe and legal, would seem best to apply for the Dominican license is the way to go.
To translate, what it is saying is that those with foreign licenses can drive for 90 days legally here. Then in (b) it says that anyone who has a foreign license can request a driver's license, without any other requisite but paying the fees for the license and presenting a certificate from the consul of the country where the license is from and with a certificate that the license is valid and with a medical certificate.
To read the law, see
http://www.ottt.gob.do/LinkClick.aspx?fileticket=RMQaUR9WHqQ=&tabid=64&mid=406