Can I still land a job with my DWI charge from last year?

I got a DWI last year. Reduced to obstruction, but what the heck, it doesn't make a difference, I just did a background check and it says charged with DWI convicted of obstruction (what is the point of that?!). Anyway, I’m in a treatment court for veterans. I was an alcoholic in the military and was released under honorable conditions, and got a DWI out of the military. Working on degree in college. Letter of recommendation from court house saying I completed the program and have been rehabilitated. Chances of landing job with honesty? (Company has 3 year DWI window I asked them)