Is a woman who used to give charity sinful if she withholds or reduces her charitable giving after becoming poor, fearing for herself and her children's ruin? And what about the Hadith: "Do not withhold, lest Allah withhold from you"? How should charity be given in times of poverty?

As long as you did not vow those charities, they are not obligatory upon you, and you do not sin by not giving them. The charity that is obligatory by Sharia (definition) is zakat (definition). As for voluntary charity, it does not become obligatory merely by habitually giving it. The poor person may give charity from what is surplus to his expenses and the expenses of those he supports, and he should not neglect the obligatory expenses upon him. Jurists have explicitly stated the prohibition of giving charity for one who is in need of what he gives in charity for his own sufficiency or the sufficiency of those whose maintenance is incumbent upon him, or for paying off his debt. The best charity is that which is given after fulfilling the rights of oneself and one's dependents, such that the giver does not become needy after his charity. The hadith (definition) of Asma, "Give, and do not tie up [your purse], lest it be tied up on you," does not mean that Asma was commanded to give charity despite her poverty; rather, it is an encouragement to give charity from what her husband brings, and it does not contradict the principle that the giver starts with himself and gives charity from what is surplus.