Well, it's less a matter of society in general being too sensitive, more a very vocal minority that gets a lot of media attention because scandal and sensationalism get page views. There are absolutely still a lot of sexist, racist, homophobic, etc. things out there, and pointing those out is not a matter of being too sensitive. Kids putting on blackface for a laugh? Yeah, it's good to educate them on the historical context of that and why it's offensive to a lot of people, so at least they'll know they're being a shit if they decide to keep doing it. Getting mad at people using certain words and slang because "cultural appropriation" is horrible and wrong? Nah, that's silly, there always has been and always will be some swapping and exchanging of language between cultures and subcultures, and it's almost never a matter of purposely trying to be shits to some other group by taking their words.
The too sensitive thing becomes a problem when silly people impede their own goals and progressive movements by raging about nonsense. For instance, Assassin's Creed Unity got a lot of crap for not having any female playable characters, especially because they gave a crap excuse along the lines of "it's way too much work to do a female model and make it work" despite having plenty of female models already in the game that move just fine (when the game isn't being a buggy broken piece of shit at least). Complaining about that made total sense. However, they recently announced that Assassin's Creed Syndicate will have a playable female character, because the main characters will be a pair of twins, and they'll have differing play styles rather than the woman just being a clone of the guy. That caused a lot of rage because, from what I saw, people were saying that Ubisoft was just pandering to feminists and didn't really give a shit about having female characters and it's somehow awful and sexist that they announced the female character thing at all. That kind of thing makes people look at the whole situation as a lose-lose, can't ever get away from the raging anger of internet activists, so why bother trying to give them what they want anyway? It hinders their own cause, but they're too busy competing with each other to see who can be the most morally outraged to realize it.
It's also a huge problem in universities. A university is supposed to be a place that challenges your thinking, that forces you to broaden your mind and consider viewpoints you'd never given a chance, that teaches you things beyond your own small horizon of knowledge and interest. Instead, hypersensitive people demand that uncomfortable topics (everything from Greek mythology in classic literature classes to rape law in a law school) be removed or labeled with a trigger warning so students can opt out of it and not learn those things at all. Their argument that such subjects can't be broached because "they might trigger memories of trauma in people and that's bad" is nonsensical, because if you ask any credible psychiatrist they'll tell you that exposure is the best way for someone to get over trauma, and coddling and avoidance are how people end up taking years and years to get over something. It's both counter to the purpose of universities and harmful to the mental health of the students, and it really needs to stop.
Basically there are points at which reasonable concern stops and oversensitivity begins, and by my observation it tends to be balanced on when that sensitivity becomes counterproductive.
