Desdemona is far from being the quiet, obedient, even submissive daughter and wife as she's often described.
Desdemona has no fear of disappointing her father, Brabantio, or going against familial and societal conventions and expectations by marrying Othello.
Desdemona loves Othello unreservedly. In her first appearance in the play, in act 1, scene 3, she has no fear of speaking up to her father, the Duke, and the council of Senators in defense of Othello and her marriage to him. Desdemona is polite and respectful to her father and the council, but she's also clearly determined to remain with Othello.
In act 2, scene 1, she plays words with Iago like a soldier. In act 3, scene 3, she's not at all hesitant to speak to Othello on Cassio's behalf, and even to risk Othello's displeasure by insisting on arranging a time that Othello will meet with Cassio.
In act 4, scene 1, when Othello strikes her in a jealous rage, she doesn't cower in front of Othello and the others in the scene, but firmly asserts "I have not deserved this" (4.1.256).
In act 4, scene 2, she seems submissive, but she's actually confused and dismayed by Othello's behavior, and she does her best to speak up for herself.
DESDEMONA. Upon my knees, what doth your speech import?
I understand a fury in your words,
But not the words.
OTHELLO. Why, what art thou?
DESDEMONA. Your wife, my lord, your true and loyal wife....
OTHELLO: Heaven truly knows that thou art false as hell.
DESDEMONA. To whom, my lord? with whom? how am I false?
OTHELLO. Impudent strumpet!
DESDEMONA. By heaven, you do me wrong.
OTHELLO. Are not you a strumpet?
DESDEMONA. No, as I am a Christian.
If to preserve this vessel for my lord
From any other foul unlawful touch
Be not to be a strumpet, I am none (4.2.35–39, 45–46, 89–95).
In act 5, scene 2, Desdemona is clearly fearing for her life, but she nevertheless defends herself to Othello.
DESDEMONA. ...I never did
Offend you in my life; never loved Cassio
But with such general warranty of heaven
As I might love. I never gave him token (5.2.68–71).
With her dying words she proclaims her innocence, but also relieves Othello of the responsibility for her death.
DESDEMONA. A guiltless death I die.
EMILIA. O, who hath done this deed?
DESDEMONA. Nobody; I myself. Farewell;
Commend me to my kind lord. O, farewell! (5.2.146–149)
Desdemona is saying either that she killed herself, or that she's responsible for her own death, although she still doesn't understand the reason for Othello's jealous rage.
Her final words convey kindness and love.
Emilia is complicit in Iago's schemes, even if unknowingly—which is a matter for another discussion. Emelia might have loved her husband in the past, and might love him now, but the emotion governing her actions is fear of Iago, although she seems to believe that she voluntarily acquiesces to his wants and needs.
EMILIA. I am glad I have found this napkin [Desdemona's handkerchief from Othello]:
This was her first remembrance from the Moor:
My wayward husband hath a hundred times
Woo'd me to steal it; but she so loves the token,
For he conjured her she should ever keep it,
That she reserves it evermore about her
To kiss and talk to. I'll have the work ta'en out,
And give't Iago. What he will do with it
Heaven knows, not I;
I nothing but to please his fantasy (3.3.323–332).
In act 4, scene 3, Desdemona finds it hard to believe that any woman would be unfaithful to her husband. Emilia seems to justify infidelity, and also rationalizes what she does on Iago's behalf.
DESDEMONA. ...Dost thou in conscience think—tell me, Emilia—That there be women do abuse their husbands
In such gross kind?
EMILIA. There be some such, no question....
DESDEMONA. I do not think there is any such woman.
EMILIA: ...Yes, a dozen, and as many to the vantage as would
store the world they played for.
But I do think it is their husbands' faults
If wives do fall; ... What is it that they do
When they change us for others? Is it sport?
I think it is. And doth affection breed it?
I think it doth. Is't frailty that thus errs?
It is so too. And have not we affections,
Desires for sport, and frailty, as men have?
Then let them use us well; else let them know,
The ills we do, their ills instruct us so (4.3.63–67, 88–108).
If Emelia has, in fact, been unfaithful to Iago with Othello, as Iago suspects (1.3.397–400), this might be reason enough to keep Iago happy, and be complicit in his schemes.
Throughout the play, Emelia has no illusions about who and what Iago is. She tries to redeem herself at the end of the play when she realizes that Desdemona is dead because she stole the handkerchief from her—which she's already tried to excuse as simply indulging her husband's whim.
Iago stabs Emilia when she tells Othello the truth about the handkerchief and Iago's plot. Emilia believes that by confessing her unwitting participation in the plot that she's relieved of the responsibility for her part in Desdemona's death.
EMELIA. ...Moor, she was chaste; she loved thee, cruel Moor;
So come my soul to bliss, as I speak true;
So speaking as I think, I die, I die (5.2.292–294).
Unfortunately, Emilia's confession comes too late for Desdemona and for herself, and it hardly suffices to exonerate Emelia from her complicity in Iago's plot to destroy Othello and Desdemona.