Which legal obligations that it's competitors don't have? If you're talking about the requirement to pre-fund their pension fund, my understanding is that UPS and FedEx (and all private corporations) were already required to do so. I think it's more accurate to say that PAEA removed an artificial advantage that USPS had.
OTOH, FedEx responded to MAP-21 and similar pension funding laws by getting rid of their pension fund which I suppose isn't something USPS can do as easily. On the third hand, UPS is still able to afford pensions for union employees.