Is it obligatory for me to abandon something I promised an elderly, devout person I would abandon, even though I do not wish to do so in my heart? Is this considered a promise or a trust? What are the consequences of not fulfilling this promise?

The answer depends on the nature of the thing promised to be abandoned, and whether it is legally forbidden (haram (definition)) or genuinely harmful. If it is forbidden by a religious text or causes verifiable harm, it must be abandoned immediately, because Allah took a covenant from the Children of Adam for their obedience, and due to the general religious evidences prohibiting harm. As for the saying "Insha'Allah" (God willing), it is not a firm promise, and nothing becomes obligatory by it except with intention. So, if the thing is not a sin or an aggression, there is no blame upon you for doing it.