What is the solution for someone who shames and curses others publicly, and does keeping silent in such a situation diminish one's dignity?

The correct solution when someone insults, slanders, and belittles you is to turn away from them. Allah Almighty says: {And when the ignorant address them [harshly], they say [only], "Peace."} (25:63), and He Almighty says: {Take what is given freely, enjoin what is good, and turn away from the ignorant.} (7:199). The Prophet (peace be upon him), his companions, and the righteous after them acted upon this noble character. For example, Umar (may Allah be pleased with him) became angry with Uyaynah ibn Hisn when he showed disrespect. But when Al-Hurr ibn Qays reminded him of the verse: {Take what is given freely, enjoin what is good, and turn away from the ignorant,} and that this man was among the ignorant, Umar did not go beyond it. The Prophet (peace be upon him) said: "If a person insults you and slanders you with what he knows about you, do not slander him with what you know about him, for the evil consequences of that are upon him." And the poet said:

When the foolish one speaks, do not answer him,

For silence is better than responding to him.

And it was also said:

When the foolish one speaks, do not answer him,

For silence is better than responding to him.

For if you answer him, you relieve him,

But if you leave him, he dies of anguish.

And it is attributed to Al-Shafi'i:

The foolish one addresses me with every ugliness,

But I dislike being a responder to him.

He increases in foolishness, and I increase in forbearance,

Like oud, the more it is burned, the better its fragrance.

So, emulate the conduct of these righteous ones and turn away from the ignorant.