I am a Romanian citizen married to a Dominican citizen. We would like to travel to Romania but there is no embassy in the Dominican Republic. The embassy of Venezuela informed me that they don't deal with visas for Dominicans married to Romanians… and the embassy of Colombia never seem to answer the phone or any of the emails I sent. This is very frustrating as we are trying to find out something for two months now.
According to this page, the Romanian embassy in Colombia is indeed responsible for the consular services for Dominican Republic.
In addition, the following information found on the site of Romanian ministry of Foreign Affairs seems relevant for you:
If you can get a free long-stay multiple-entry visa for your husband, that seems like the best option for you.