You can look up your DC case record online (by name or case number) using the following link: https://www.dccourts.gov/pa/
The online docket should reflect the current case status, and whether the Court has issued a bench warrant. I would suggest hiring a local lawyer to help you contact the Court, handle the potential consequences and eventually complete of your sentence. You may want to call your original defense attorney. Another good resource is the DC Public Defender Service: http://www.pdsdc.org (they have a lawyer on duty to answer questions during business hours).
A criminal record for an OWI will not prevent you from obtaining a passport, but I would be sure to address any unresolved issues with your 2005 case, especially a warrant, before applying. As for "clearing your name," you can file to seal a criminal record in DC only if the charge and its final disposition satisfy the requirements of the DC Record Sealing Act (unless you can prove that you were actually innocent). Even if eligible, you must wait to file once you have satisfied all sentencing requirements set by the Court and then a waiting period. The facts that you recited above seem to indicate that you were convicted of Operating While Impaired. Unfortunately, a conviction for OWI cannot be sealing under current DC law as it is an "ineligible misdemeanor." To be eligible to file a Motion to Seal an OWI, the charge must have been somehow dismissed (e.g. a no paper, nolle prossequi, acquittal, dismissal or conviction set-aside) and the defendant waited five years after completing all sentence requirements.
I hope this information proves helpful. Best of luck.