What are at least two examples in the text where you feel Douglass is exaggerating or self-glorifying?

Life’s greatest heroes are imperfect. In fiction, it’s often the flaws that humanize our heroes. In nonfiction, it’s the responsibility of the author to be reliable—to factually recount both the good and the bad. It’s difficult to write truthfully, especially in an autobiography, about mistakes one has made or personal failings of character. The author may wish to tell a story a little differently than it really happened, to self-aggrandize or edit the dialogue to appear a little sharper, kinder, or more capable. Readers expect autobiographies to describe what really happened, but most know it wouldn’t be possible to be exactly right about every detail. Most people don’t record every conversation, and the human mind is subject to misremembering an event or even developing an entirely new perspective over time.
Frederick Douglass, a heroic figure, is surely subject to the same flaws as any other man. He revised and republished his narrative multiple times, each with differences in the way some events transpire. The differences, however, are not in substance so much as in tone. For example, in chapter 10 of the first edition of his autobiography, Narrative of the Life of Frederick Douglass (1845), he writes:

I seized Covey hard by the throat; and as I did so, I rose. He held on to me, and I to him. My resistance was so entirely unexpected that Covey seemed taken all aback. He trembled like a leaf. This gave me assurance, and I held him uneasy, causing the blood to run where I touched him with the ends of my fingers. . . . He meant to knock me down. But just as he was leaning over to get the stick, I seized him with both hands by his collar, and brought him by a sudden snatch to the ground.

The same event, recounted in chapter 17 of the 1855 version, called My Bondage and My Freedom, reads:

The fighting madness had come upon me, and I found my strong fingers firmly attached to the throat of my cowardly tormentor; as heedless of consequences, at the moment, as though we stood as equals before the law. The very color of the man was forgotten. I felt as supple as a cat, and was ready for the snakish creature at every turn. Every blow of his was parried, though I dealt no blows in turn. I was strictly on the defensive, preventing him from injuring me, rather than trying to injure him. I flung him on the ground several times . . .

This revision says the same thing, effectively, but with a more heavy-handed writing style. While it could be seen as self-aggrandizing, Douglass does not come off as more impressive in the latter version. In fact, he loses a bit of his power in the extra details.
Some people suggest that Douglass, as a man enslaved in Maryland, exaggerates the hardships of slavery. Maryland is not the Deep South, after all. Critics believe in particular that Mrs. Auld, who teaches Douglass the alphabet, is unfairly treated in his characterizations of her, as in chapter 6 of his Narrative:

Her face was made of heavenly smiles, and her voice of tranquil music.
But, alas! this kind heart had but a short time to remain such. The fatal poison of irresponsible power was already in her hands, and soon commenced its infernal work. That cheerful eye, under the influence of slavery, soon became red with rage; that voice, made all of sweet accord, changed to one of harsh and horrid discord; and that angelic face gave place to that of a demon.

That Douglass exaggerated the horrors of slavery is not an advisable argument to make. Doing so is effectively to say that Douglass should have a more charitable perspective of a terrible institution. While it is entirely appropriate to argue that Douglass's revisions add unnecessary embellishments, it’s difficult to find selections where he clearly exaggerates outright or seeks to glorify himself.
Note: See the attached link for a letter written in 1845 by one of Douglass's early critics. He defends not only the slave masters but also the institution of slavery.
https://docsouth.unc.edu/neh/douglass/support2.html