The Dominican Republic has set up a page, in English, that guides you through the process of extending your stay in the country for up to 4 months. It can all be done online.
migracion.gob.do
STAYING EXTENSION - DIRECCIÓN GENERAL DE MIGRACIÓN
Authorize foreigners interested in prolonging their stay in the Dominican Republic for more than 30, through an extension that can be extended up to 120 days.