I just read on another sub an unbelievable story about a sniper during the Vietnam War who performed multiple incredible feats and was honored for it. While I was in the Air Force we learned of many heroic war stories like these, about snipers or pilots or infantry men.
It always struck me that there is significant incentive for these war stories to be exaggerated for the purposes of propaganda and boosting morale.
Is the methodology used to verify these war stories trust worthy? Are most famous stories bunk? Is it safe to assume that the military wants the stories of genuine heroic feats to become well known? How do historians deal with the issue of veracity?
It very much depends on each individual case, and what can the corroborating external evidence. As you can see from my flair, I specialise in early modern Spain, a period with some interesting war stories. I will give you one that can be relevant to the question.
In the year 1525, on February the 24th, there was a very important battle, the battle of Pavia, that resulted in a crushing defeat for the French army: the King of France himself was captured, and so were many high ranking French people, and many more were killed. Capturing the French king is a feat that would result in the captors being promoted and rewarded, and so they were.
Francis I was taken prisoner by Alonso Pita da Veiga, Juan de Urbieta, and Diego Dávila. How do we know that? There are some accounts of the battle, like the one written by Juan de Oznayo, who was a page to the Spanish general Marquess of Vasto. He describes the effort by those three men at arms, but he was not a direct witness of the capture. Alonso Pita da Veiga himself wrote a summarial account of the capture of the King in that battle. He tells that he, Dávila, and Urbieta ganged up on the French king, with Dávila grabbing Francis by the arm and taking his sword and right gauntlet, Urbieta taking the King's horse's reins and the King's targe, and Pita himself blocking the way for the king, taking the left gauntlet and the sash.
Francis I of France personally wrote three notes, one for each of the captors, attesting their bravery and honourable conduct in taking him prisoner. There is a notarial copy of Urbieta's letter, and Pita's original still survives in the possession of the Pita da Veiga family in their manor of Mandiá. We also have the royal charters giving these men promotions and prizes. Dávila sold the sword and the gauntlet to the Crown. Besides all of this, there are letters by the Count of Salm, one of the generals of the imperial army that day, attesting the feats and recommending the men for rewards.
The strong coherence between Pita's account, the surviving documents, and the presence of the sword, a gauntlet, and the targe, all can attest to the veracity of the action as performed by the three men at arms. Of course, not all war tales can be verified to that thorough level of scrutiny.
Given what you're asking about, this older answer might be of interest.