The "truth" is already out there, in media reports of the time, in official NASA briefs, in a number of books and documentaries, even the Ron Howard movie is fairly accurate (although they did overplay the Ken Mattingly/Jack Swigart friction).
I highly recommend:
"Lost Moon" by Kluger and Lovell
"Failure Is Not An Option" by Gene Kranz
"Moon Shot" by Shepard, Slayton, Barbree and Armstrong
and although I haven't read it, "Apollo" by Murray and Cox is back in print and highly regarded.
Overall, though, I personally hate the meme "Apollo 13 was a disaster" and "was saved." Not to downplay the explosion and the potential danger, it was the astronauts, the ground crew, the engineers, the builders who overcame a huge problem by being smart, cautious and prepared enough for the unexpected. As Gene Kranz said and was repeated in dramatic fashion in the movie - "This is our greatest triumph."
By the way, did everyone know that, for the last 42 years, the records for "fastest speed achieved in human flight" and "farthest distance humans have traveled away from Earth" is held by the Apollo 13 astronauts? That high lunar orbit they pulled off got them those records.
Not bad for a so-called "failure."