Is one sinful for refraining from donating money to help urgent medical cases circulating on social media, especially if the money saved is for one’s basic needs or to buy things that exceed them, and is one obligated to publicize these cases to others?

There is no sin upon you for what you mentioned regarding not giving away all your money in charity or not informing others about the cases. Rather, you are a doer of good and will be rewarded for donating blood and money. It is not obligatory to give away all one's money in charity; indeed, the Prophet, peace and blessings be upon him, forbade Ka'b ibn Malik, may Allah be pleased with him, from doing so. Informing others about cases and publicizing them is good and a praiseworthy endeavor, but it is not obligatory unless a definite harm would result to the patient from not announcing it.