What is to be done if performing the Hajj obligation on behalf of a deceased paternal aunt is not possible due to financial circumstances and the decreasing value of jewelry designated for that purpose, given that annual zakat is paid on it?

Your being an employee is not an impediment to performing the obligatory Hajj; rather, it should help you cover its costs. Hajj is obligatory immediately for those who are able, and it is not permissible to delay it. The Prophet, peace and blessings be upon him, explained ability as possessing provisions and a ride, and that this should be in excess of the expenses for those whose maintenance is obligatory upon you. If you are able, then you must first perform the obligatory Hajj, and then execute your aunt's will. As for the zakat (definition) on the mentioned gold, its disbursement is incorrect because the wealth did not have a specific owner before the incapacitation, and afterward, it became the property of the heirs. It is not permissible for one who has not performed the obligatory Hajj (Hajjat al-Islam) to perform Hajj on behalf of another, according to the preponderant opinion. If your aunt passed away after she was able to perform Hajj but did not, and she left behind wealth, it is obligatory for her heirs to pay for someone to perform Hajj on her behalf from her wealth.