Getting Back on Your Hooves has an example for Trixie. After going on an adventure in Everfree Forest with Fluttershy, in which she saw that her belief that kindness equals weakness was very flawed, she sings "Through New Eyes", about how her world view has changed and she's not certain what to do now. Twilight joins in, reassuring Trixie with her own part of the song about how her world view was changed by her move to Ponyville. Trixie finishes by singing about how she now knows her new True Companion Twilight will help her through the change.
Film - Animated
The Last Unicorn: "Now That I'm A Woman"; in this case while the physical change from unicorn to woman has already taken place, Amalthea is describing the mental changes that are taking her over.
"Strange Things (Are Happening to Me)" from Toy Story counts, since Andy starts to appear to favor Buzz over Woody, and this has an effect on Woody's character.
"I am Chirin" from Chirinno Suzu, which is played when Chirin decides to team up with the evil Wolf King, and as a result he goes from an innocent baby lamb to a fearsome, demonic ram.
Frozen: Elsa sings "Let It Go" when she goes into a self-imposed exile after her powers are exposed in public, and now she can use her powers to her full potential and build an ice palace in total solitude.
Film - Live Action
Eric Draven gets one in The Crow (specifically, "Burn," by The Cure) as he goes to a makeup mirror in his apartment and makes his transformation into a white-faced, spandex-clad avenger.
Hannah Montana: The Movie gives us "The Climb", which neatly encapsulates Miley's emotions, the lessons she's learned in the movie, and the general message of the film. Later, in Season 4, Miley writes "Wherever I Go" to say goodbye to her secret identity as Hannah, and moves forward to face the world as her real self.
The Muppets has "Man or Muppet" in which Gary and Walter both realise they need to move beyond their sibling relationship (Gary by actually proposing to Mary, Walter by joining the Muppets).
In Wicked, after Elphaba finds out that what she assumed was the ultimate force for good—the Wizard—isn't, she knows she has to help the Animals without him. 'Defying Gravity' echoes her "I Want" Song, 'The Wizard And I', now with her asking for Glinda's partnership instead of the Wizard's.
"What Have I Done" is Valjean rejecting his life as a criminal in favor of redemption.
Cosette's version of "In My Life" combines a grown-up "I Want" Song with I Am Becoming Song. She sings about her longing for new horizons and love, and closure with the past - and how she's growing up and delighted to do so.
In The Phantom of the Opera, "Wishing You Were Somehow Here Again," Christine's lament for her father, ends with a declaration that she will try to put his memory behind her and live unburdened by the pain of her past.
Carmen has "In L.A." in the musical version of Fame, which is about how she moved to L.A., got hooked on drugs, and slept with producers for money/drugs. Doubles as a BSOD Song.
Thirteen has "13/Becoming a Man" which is about the main character's Bar Mitzvah.
"La Monture" ("My Heart If You Will Swear") from Notre-Dame de Paris: the devoted, virginal Fleur-de-Lys tells her cheating fiancÚ Phoebus that she will remain faithful to him... as long as he has Esmeralda executed. In the French lyrics, she sings that she has thrown her girlhood dreams "to the wolves" and that her heart is "hardening."
In "An Organized Life (1974)" from the musical version of Vanities, Kathy sings about her nervous breakdown and journey of self-discovery, also making it a BSOD Song. Also, "Friendship Isn't What it Used to Be" from the same act, and "Looking Good", the finale for the TheatreWorks Palo Alto and off-Broadway versions.
In Dreamgirls, Effie sings "I Am Changing" and decides to get her life together. In the revival of the musical, the actress playing her has changed into a glitzy dress that is revealed just before she starts the final belt.
Elle has several in Legally Blonde: The Musical from "Chip On My Shoulder" to the reprise of "Legally Blonde". "So Much Better" in particular stands out as almost being this play's answer to "Defying Gravity" in terms of codifying Elle's metamorphosis.
In Little Shop of Horrors, Audrey's part in the duet "Suddenly Seymour", is about how she believes that Seymour is making her a better person.