Is it obligatory to remain silent about my aunt's transgression of cursing her mother and going out with men, so as not to cause problems that might lead to the death of my ailing grandmother, or is it obligatory to act upon what Allah has commanded and not remain silent about transgression?

Enjoining good and forbidding evil is a great ritual in Islam. It is obligatory upon a Muslim according to their capacity and the prescribed Islamic guidelines, provided that forbidding evil does not lead to a greater evil, that the one forbidding has the ability to change, and that they do not expose themselves to an unbearable tribulation. Regarding your paternal aunt, pray for her guidance, and continue to advise her gently and kindly, taking into account her psychological state. If possible, change your place of residence to cut off her relationships. If she does not desist, let the family members take a unified stance to prevent her from going out and meeting strangers, and let one of her mahram accompany her if absolutely necessary. If none of this happens, then you have fulfilled your duty and there is no sin upon you.