it's a little odd that you would have breakage issues given that your hair is 3-5 inches; usually breakage doesn't kick in until you're dealing with shoulder length tresses or longer.
which leads me to wonder how gentle are you when you detangle? You describe your hair as coarse, but are you referring to your strands being bigger than average or do you mean coarse in terms of feel (hair feels rough or abrasive to the touch) because if it's the latter, that can be remedied through moisture and finding the right products (or if your hair's like mine, then using less)
the other thing is i would wonder if your breakage might be more your perception of your hair breaking. what i mean by that is that your hair is not breaking any more than what is normal and can be expected for kinky coily/curly hair types, but that because we want to eliminate all of it we feel like we're not doing right by our hair when really we're doing the best we can do. we over think it and over exaggerate the damage factor. if you're only breaking about 10 or so strands (how many would you say you're losing to just breakage?) and you only detangle your hair on wash days with a sulfate shampoo, then that's really not a bad trade-off and i'd say you're doing a great job!
But if your breakage is worse than this, there are a few explanations:
1) Your hair doesn't care for wet detangling and would prefer dry oil detangling instead
2) your shampoo is roughening your hair, increasing its susceptibility to breakage
3) your hair is not adequately detangled prior to washing it, so washing it in a tangled state only exacerbates the issue
4) you're not being gentle/patient enough with your hair when you detangle (not sure if you finger detangle or not but i recommend it if you don't already do it, because it's much more gentle than using a comb once you get used to it)
i think number four is the most likely choice, since everything else in your routine sounds solid. :)