As I think you suspect, the offensiveness of "the N word" has indeed increased.
It may always have been that offensive to black folk. The difference is that back in Twain's day, white folk didn't really care how they felt about it. Today they (for the most part) do.
I'm actually old enough to have personally seen a certain amount of this transition. Until about the mid 70's, white folk had very little problem saying the "N-word". It was considered derogatory, but only in the way similar words like "Jew" are. I went to an integrated school, and used to hear it regularly when someone got upset with a black schoolmate.
In the mid-70's things started to change. I think I heard the word from the lips of a white person exactly once (and in a whisper) between 1977 and 1980, and never since then.
This process can be seen in popular culture. Mel Brooks made a hilarious movie satirizing race relations in 1974 called Blazing Saddles. As such, naturally the evil or ignorant white folk in the movie casually used the "N-word" throughout directly at the black protagonist. Today it is very difficult to find a copy of that movie that doesn't alter the dialog, and watching such a copy is very uncomfortable.
Three years later another comedy named Kentucky Fried Movie was released. This was right when the transition was happening, and a skit in there captures it perfectly. It was meant to satirize the change in the acceptability of the word by showing a white person commiting suicide by merely walking into a rough neighborhood and shouting the word. When watched today, it doesn't seem funny at all, and one's first reaction is that the jerk deserves whatever he gets. But clearly things with that word were different (and in transition) when it was written, because somebody thought it funny. The joke is just "dated" to a time when everybody remembers the word being more acceptable.