All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
First of all, you should know that it is not permissible to accuse a Muslim of committing fornication or adultery except with evidence. This is even more forbidden in regard to one’s mother because of her great status. Allaah says (what means): {O you who have believed, avoid much [negative] assumption. Indeed, some assumption is sin.} [Quran 49:12]
In case this is proven about her, then she must be advised and called to repent. If she repents to Allaah, then the matter is clear. In case she does not repent, then this does not justify for her children to abandon her or mistreat her. Indeed, Allaah ordered us to be kind and dutiful to our parents even if they are disbelievers and trying to mislead their children. Allaah says (what means): {But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them but accompany them in [this] world with appropriate kindness.} [Quran 31:15]
The fact that she committed evil does not exempt her children from being kind and dutiful to her. Al-Bukhari wrote a chapter with the title: Being kind and dutiful to the parents even if they are unjust (or wrong) in his book Al-Adab Al-Mufrad. In this chapter, he quoted a statement by Ibn Abbaas who said, "There is no Muslim who has two Muslim parents and he serves them seeking the pleasure of Allaah, except that Allaah opens for him two doors, meaning to Paradise, and if it was one parent, then Allaah will open for him one door (to Paradise), and if he angers one of them, then Allaah will not be pleased with him until his parent is pleased with him. He was asked, 'What if they wronged him?' He replied, 'Even if they wronged him.'"
Hence, this act that you mentioned about your brothers, that they do not respect her and insult her with bad names, is a great evil, and it is undutifulness towards her, so they must be advised to stop doing so.
For more benefit, please refer to fataawa 87019, 82254 and 87856, which are about the virtues of honoring one's parents and treating them kindly and the prohibition of showing undutifulness towards them.
Also, it is forbidden to leave your mother in a condition that may lead her to ruin, and if she has no money, then it is an obligation on all her children, males and females, to spend on her, each according to their ability.
On the other hand, your mother has no right to oblige you to marry a specific man or to oblige you to live with her. You should choose a religious man who has a good moral character as a husband regardless of whether he is your cousin or anyone else, because such a husband will not hold you accountable for the sins of your mother, so you should repel any bad thoughts in this regard.
Allaah knows best.