Well, why didn't you say so at the outset?
I consulted the online MO State Archives military service records from the MO Sec'y of State's website and found that three Washburns from Lewis County joined Porter all on the same day. Argile, Benjamin, and S. D. (the clerk may have been dyslexic) Washburn all joined Company B, 2nd Northeast Missouri Cavalry Regiment of Colonel Joseph Crisman Porter's NE MO force all on 25 July 1862, place not stated. S. D. Washburn's record also states "Shot by Meril at Huntsville, Mo. Date not on record." So there is no doubt this S. D. Washburn was your D. S. Washburn.
There is no record of any prior service, but the southern records in this collection are notoriously incomplete. Some on this forum would quip that is what happens to the losing side of a war. I can't argue with that logic.
The point is that a southern man did not have to serve on the Rebel side to draw oath of bond from the Federals. That was also imposed many times simply because a particular citizen was a "notorious Rebel" as stated in Missouri Union military regulations. If Washburn had not served in the southern Missouri State Guard in 1861, perhaps he was outspoken in his Lewis County, was seen in public making secession utterances, assisted local southern units to recruit, brought food and supplies to local southern units, harassed northern sympathizers, and so forth. I rather doubt voting records had much to do with this, unless Washburn was seen to harass northern sympathizers at the polls. Any of these charges were sufficient in northeast MO in 1862 to draw oath and bond (or worse) from Union authorities. Sadly, we don't know the specific cause of why oath and bond were imposed on D. S. Washburn, so this is all conjecture.
The "O.R." reference you mentioned is "O.R." series 1, volume 13, page 660. This lists General Lewis Merrill, formerly commander of 2nd Missouri Cavalry ("Merrill's Horse"), as of the time of the execution order in September 1862 as occupying "Headquarterd Northeast Missouri Division." That headquarters at Macon, Macon County, would presumably have at its disposal all the northeast Missouri lists of "notorious Rebels," oath takers, and bond payers, and other such lists, compliments of the provisional governor's office and the U.S. Army. Further, Merrill would have at his disposal lots of reports from informants, detectives, turncoats, neighbors, co-workers, and the like. After all, this was a civil war--neighbor against neighbor, and all that. Nasty business. That is how the Union military probably knew D. S. Washburn was present with members of Porter's command at the Sugar Camp on 28 July 1862 in violation of the oath he had taken in March that he would not bear arms against the U.S. government. (See earlier discussions in this forum about such oaths in MO. They nearly all said the same thing.)
Some oath violators taken by Union troops during the Porter campaign in northeast Missouri were shot in the field, like after the Kirksville battle. These men were captured with their oath paper on their person. In the case of Washburn, the Union military chose to investigate and make a case before they shot him.
I hope that helps.