Am I obligated to pay off my brother's debts after I pledged to do so, especially since he caused me embarrassment, knowing that he is employed and capable of paying them off?

If the pledge to repay the debt is understood conventionally as a guarantee, then you become a guarantor. A guarantee is valid with any expression from which a guarantee is understood conventionally. The guarantee is a binding contract from the guarantor's side, and it cannot be revoked except by the absolution of those for whom the guarantee was made. The beneficiary of the guarantee has the right to demand the debt from the guarantor, even if the person for whom the guarantee was made is able to repay. However, if the pledge is merely a promise that does not conventionally imply a guarantee, then the promise is not binding, and fulfilling it is not obligatory according to the majority of scholars; rather, it is merely recommended.