Just a theory
Haven't seen this movie yet, so only have read about this case.
There's a line in the song Tourettes on In Utero album - "One more year to go of my life"
There's the emotional performance of "In the Pines" on Unplugged.
The entire In Utero album comes across as the work of someone in extreme emotional inner turmoil, more so than their other albums.
Anyway - my theory is the guy was ashamed at being a junkie and a father, horrified at the thought of his marriage failing, because of how divorce impacted him, and just couldn't deal with life anymore.
He wrote the first part of the letter, shot himself up with a giant dose of heroin hoping it would finish him off, but it didn't, then he added the part in the bad handwriting at the end of his letter and thats why it looks different, he then decided to finish the job with the shotgun. Kind of how HITLER took a cyanide capsule and shot himself at the same time.
He probably had a tolerance to the high dose of heroin from years of abuse.
Just my theory from what I know about Kurt, Nirvana, and the case.
He probably had a high tolerance