First the disclaimer, I am not licensed in DC and I have no idea where you got your DWI case and/or the laws of that jurisdiction (unless it was in Texas). This is not legal advice and I am not your attorney.
My initial reaction is "Why not apply?" Really, what's the worst thing that could happen? They reject your application and you don't get the job. You have a 0% chance of getting the job if you don't apply.
Don't let one stupid DWI ruin your life. George W. Bush had a DWI and they still let him be President, Dick Cheney had two in his party days. DWI isn't murder or theft, it happens to good people all the time.
Finally, consult with a local attorney to see if your record can be cleared or expunged.