I think this will answer most of your questions. Col. Willoughby Babcock of the 75th New York gives the best writeup I have seen of this particular incident. Babcock's take on this comes from his personal diary located at the Minnesota Historical Society. It is worth the read. We have a copy of a portion of it at Camp Moore. Keep in mind that Babcock is keeping a diary here and that this is not a memoir some years after the war. Babcock would not survive the war. Here is what Babcock's take on this was:

"On the day after this execution (6th), one Richard H. Glaze was arrested by Dwight's cavalry, approaching the column, mounted, with overcoat and blanket, coiled and strapped to his saddle, and armed with a heavy hunting rifle which was loaded with ball. He declared on being questioned that he had once belonged to the 8th La. Regt. and had been discharged for disability, and that he did not now belong to any military organization. He was taken before Gen. Dwight, and on his own statement and appearance was ordered by Gen. Dwight, without trial, to be shot, and he was accordingly shot on the morning of the 7th May within miles of his own home. The fact was that he was a member of Capt. Murdoch's Company of Cavalry, a company of mounted militia raised for home defense. Glaze had been left behind on account of a debauch at the time of their flight and was on his way to overtake his company when arrested. This information was derived by me from his neighbors, many of whom I saw on Bayou Huffpower."

"The pretext for this hasty severity was the death of Captain Howard Dwight, a brother of Gen. Dwight, on the Monday previous, at the hands of some mounted prowlers who had come out of the fields on the opposite side of Bayou Boeuf, some 6 or 8 miles from Washington, and attempted to capture Capt. Dwight who was riding alone about half a mile behind the wagon train of his brigade, endeavoring to overtake the head of the column. According to a negro boy who saw the occurrence, they summoned Capt. Dwight to surrender. Confessing himself in their power, he assented to it, when they ordered him to cross the bayou to them. He was endeavoring to cross, or started to cross, when one of them cried out "Shoot the God d---m son of a b---h," and he fell pierced by two balls. The rebel account of the matter is, that seeing some of our troops coming to his aid, he turned and attempted to escape, when he was shot, and that the perpetrators of it were armed soldiers in the discharge of their duty. The general opinion among us was that it was a mere guerrilla murder. Gen. Banks ordered a thorough search for the murderers on that side of the bayou, but without success, and a large number of citizens of the neighborhood were arrested and sent to New Orleans to be held as hostages for the prevention of such outrages in the future and to compel the surrender of the murderers."

As to Glaze, here is the record of his service from my own research:
Glaze, Richard H., 2nd Sergeant

Richard H. Glaze was born Nov. 12th, 1833 in Louisiana. He was married to a woman named Sarah and farming near Holmesville, LA when he enlisted at New Orleans on June 19th, 1861. Richard was absent, sick with bowel problems in Culpepper Court House, VA beginning Aug. 28th, 1861. Evidently the sickness was serious enough to warrant his discharge from the service on Sept. 21st, 1861. He came home and started farming again. It is unclear if he ever joined a Louisiana Home Guard unit but he was captured on May 6th, 1863 near Cheneyville, LA by Gen. William Dwight's command. Glaze was captured with a Kentucky Rifle, an overcoat and a bedroll. He was accused of being part of Murdoch's Independent Company of Cavalry. Dwight's brother had been killed a few days before by Confederate troops in the area. In retaliation for his brother being killed, Gen. Dwight ordered Glaze executed by five privates of the 91st New York Infantry, without trial, on the next morning, May 7th, 1863. He was murdered within a few miles of his home. Glaze's wife was awarded a Confederate Land Warrant after the war.

Sources: "Willoughby Babcock Commentaries", Minnesota Historical Society, St. Paul, MN; Special Orders No. 41, HQ, Advanced Brigade U. S. Forces, Bayou Beouf, LA, May 6, 1863; Confederate Land Warrant records, reel 1, No. 0456, LSA.

BTW, I have a copy of the Special Orders No. 41, the Execution order if you need a copy. You can email me at: cosby_w@bellsouth.net