Is it permissible to retract the statement: "By Your might and glory, my Lord, if You save me from this problem, I will wear the niqab — the jilbab — by God, I will wear it"? And is this considered a vow and an oath, or only a vow?

This statement is considered an oath, not a vow (nadhr), because it does not contain anything that indicates a commitment. The formulation of a vow is typically, "It is incumbent upon me for Allah that I do such and such," or "I owe a vow of such and such." If you retract it, you are obligated to perform an expiation for an oath (kaffarat yamin), based on the Almighty's saying: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or freeing a slave. But he who finds not [the means to do so] - then a fast of three days. That is the expiation for your oaths when you have sworn. And guard your oaths. Thus does Allah make clear to you His verses that you may be grateful."