The burden you are carrying is very heavy, and it is understandable that you feel anxious, exhausted and without hope as you try to care for your mother while facing constant conflict with your brother over the home.
That sense of hopelessness is important to name. When a person comes to believe there is no way out and nothing can be done, heaviness and depression follow. Scripture describes a man who felt he had seen affliction and that God was not listening, yet in the middle of that despair he made a deliberate choice about what to remember. He said, this I recall to my mind, therefore I have hope. It is the faithful love of the Lord that never ends and his mercies that are new every morning. You can think yourself into despair by rehearsing every threat and every unkind word, or you can renew your mind by setting it on the Lord and his character. That does not make the problems disappear, but it moves you from total despair to a place where hope can live again because God himself has not changed.
Your desire to care for your mother is a good and right desire. The Bible teaches that the family of God includes older men and women who are to be honored as fathers and mothers, and it is very clear that children and grandchildren have the first responsibility to care for widowed parents and to meet their needs. If that responsibility is refused, Scripture says it is a serious failure, worse than not holding to the faith at all. The fact that you have stepped in to watch over your mother day and night, even when it is costly and misunderstood, reflects that God given duty. The Lord sees that faithfulness.
At the same time, the question of the house brings a painful tension between what is legal and what is right before God. In the law God gave to Israel, land was to stay within families and a near kinsman could step in as a redeemer to keep the inheritance from being lost. That whole provision was meant to protect family loyalty and to keep one member from using his position to harm another. When a brother refused to fulfill that family obligation, he was seen as unfaithful to his own family. Today the civil law may say that the person whose name is on the deed has the legal right to decide what happens to the property, and agencies will often speak in those terms. Having a legal right, however, does not mean a person is using that right in a way that pleases God. God is not honored when power over a home is used for revenge, for harsh words, or to punish someone who is trying to do what is right. Moses, who had great authority, did not use his position to build up his own family at the expense of others, and that is the pattern God honors.
It is also painful when false accusations are made out of anger. Scripture warns that people sometimes file charges that are not true because they are angry about something else and want to get even. Your brother's claim about finances was investigated and nothing was found wrong, yet the desire to get even has continued. That pattern of looking for another reason to punish you, the snarky comments about your caregiving, the refusal to give a clear answer about the future, all add to your anxiety. It is not wrong to feel hurt by that. God sees both the truth about your care for your mother and the motives behind the harassment.
You have been asking God to change your brother's heart and it feels as if he is only becoming harder. God does hear your prayers, even when you do not see an immediate change in another person. A hard heart is not a sign that God has abandoned you or that he will let wrong win. It is a call to keep entrusting both yourself and your brother to the Lord, to not return evil for evil, and to keep doing what is right quietly and faithfully. Continue to keep the home as clean and orderly as you are able, keep notes of inspections and conversations, keep communication brief, calm and in writing when possible, and seek wise, local help for the legal and practical questions about housing rights, caregiving support and what would happen for your mother if you had to move. A trusted pastor and a legal aid resource for older adults can help you understand options without having to carry all the uncertainty alone. If you are able to bring your mother with you should you need to leave, that again reflects honoring her, but her medical and daily care needs and any legal considerations for her residence will need careful counsel.
For your anxiety, bring the specific fears to the Lord each day rather than letting them run without restraint. Ask him for daily mercy, for provision if a move becomes necessary, for work and housing you can afford, and for peace in your home. He promised Jacob when he called him to return to his family and his land that he would be with him, and that promise to be present with his people has not changed. Your age and limited income and the lack of answers about the future are real concerns, and God knows them fully.
Praying for you now that the Lord will guard your mind with hope, give you wisdom for each next step, provide for you and your mother, soften what is hard, and let you experience his new mercies morning by morning. May he give you strength to continue caring for your mother with love, to act with integrity before him, and to rest in the knowledge that he is your redeemer and your help, even when people around you are not.