More on the handkercheif...
Desdemona pulls out the handkercheif when Othello complains of having a headache, meaning to help him ease his pain. Her intention is to bind his head to help take the pressure off; he tells her her handky is "too small"--meaning it is little, perhaps, but also insufficient to take away his pain.
Othello's headache is from his recent conversation with Iago which has him thinking about the possibility (now probability, in his mind) of Desdemona's unfaithfulness. She doesn't know this--she thinks he is feeling unwell from being too much on the "watch" (another play on words).
The poignancy in this scene is that she loses the handkerchief in care and concern for Othello. When he says it is "too small", she drops it; she moves to pick it up, but he instructs her to leave it, and to follow him instead, which she does.
So, in a sense, it is her love, her willingness to obey and follow Othello, instead of her worldliness--in this case, concern for material goods, that puts her at risk.
Emilia finds a handkerchief that had been a courting gift given to Desdemona by Othello. Iago had been asking Emilia to steal it, but fortune intervenes and she did not have to steal it. She explains to Iago: "...she let it drop by negligence,/And to th'advantage I, being here, took 't up" (3.3.316-318)
Iago will give the purlonied hankie to his mistress Bianca. When Desdemona cannot show the gift to Othello when he asks to see it, Othello is convinced of her adultery and swears an oath of vengeance on both her and Cassio (whom he has been led to believe by Iago is the one with whom Desdemona is cheating.)