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. We ask Allaah to exalt his mention as well as that of his family and all his companions.
If your husband accused you of cheating and stealing without evidence, then he was very wrong, as in principle a Muslim is innocent. Allaah Says (what means): {O you who have believed, avoid much [negative] assumption. Indeed, some assumption is sin}[Quran 49:12].
So, we advise you to be patient and seek the reward from Allaah; you have a good example in the Prophets, may Allaah exalt their mention, and the pious people, as they were tried and tested and were patient, so Allaah raised their status. Allaah Says (what means): {Verily, Messengers were denied before you, but they were patient over [the effects of] denial, and they were harmed until Our victory came to them.}[Quran 6:34]. The mother or the believers, 'Aa'ishah was accused of committing adultery, but she was patient and Allaah revealed her innocence from above the seven heavens, and these verses will be read until the Day Judgement. So there is good in being patient. You should read the verses of the Quran and the Prophetic narrations about the excellence of patience.
Moreover, you should know that being nervous and angry does not achieve any benefit or repel any harm, rather, this may lead a person to regret and anxiety. The Prophet said: "The greatness of the extent of the reward is proportional to the greatness of the affliction. When Allaah loves some people, He afflicts them. So, those of them who are content get the reward of being content while those who get angry get the punishment of being discontent." [At-Tirmithi and Ibn Maajah]. Therefore, you should advise your husband in a nice and soft manner. The scholars and the righteous people may also advise him, and he should be reminded to fear Allaah with regards to his wife, he should also be reminded of the saying of the Prophet when he was asked about the right of one's wife on her husband: "To feed her when you eat, and clothe her when you buy clothes for yourself, and do not hit the face and do not rebuke (or bad mouth) her, and do not abandon her except within your home." [Abu Daawood]. The fact that he left the home and abandoned his wife and children in this state is contrary to religion, and it is not permissible to abandon her within the house except for a sound religious reason. Nonetheless, you should invoke Allaah earnestly to reconcile between both of you.
As regards what he has done with you in order to marry another woman, and lying to his family, then you should not preoccupy yourself with his, as this will only cause you greater harm. Nonetheless, you are allowed to accuse him of what is mentioned in the question [of what he does with the other woman] without evidence.
Finally, it should be mentioned that if you are harmed by staying with him, then you are permitted to ask for divorce, as it is not permissible in Islam to cause harm to other people.
Allaah Knows best.