So, generally there is a large lack of sources to verify this.

This is from the Essay "The Bravest of the Brave" by Brigader (Ret'd) Peter Young.

On December 7, 1815, condemned by his peers, he appeared before a firing squad at the Carrefour de l'Observatoire in the Jardin du Luxembourg. "Soldiers," he said, "when I give the command to fire, fire straight at the heart. Wait for the order. It will be my last to you. I protest against my condemnation. I have fought a hundred battles for France, and not one against her." Genearl Count Leon Rochechouart, who was in charge of the proceedings, remembered that Ney's demeanor was "noble, calm and dignified beyond reproach. The marshal gave the order to fire, and fell struck by 11 bullets. "One of the soldiers had the good taste to hit the top of the wall."

This is a certain and definite source detailing the last moments of the Marshal. This is evidence of his death, a primary source that details it. Due to a lack of evidence to explain how he fled America and the fact that the French Crown wanted to make an example of him for others.

However, one other item is of unique interest that may prove the story incorrect. Ney's primary education was going to be that of a Notary at Sarrelouis and an overseer of mines and forges, he would have become a civil servant. He rejected this life to become a hussar in the Ancien Regime. However, this education would have given him a large amount of French legal knowledge, which wouldn't detail an extensive knowledge in Latin, Ancient Greek, or mathematics. Even Napoleon, who received one of the best education available to a petite noblesse only received Latin.

Further, beyond being interested in military theory, Ney didn't have a strong passion for academics. The military was his life and that may be one of many reasons as to why he may have developed PTSD over the years.

The entire story is dubious and thus not to be trusted.