  A new proof for stability estimates for the complex Monge-Amp\`ere and
Hessian equations is given, which does not require pluripotential theory. A
major advantage is that the resulting stability estimates are then uniform
under general degenerations of the background metric in the case of the
Monge-Amp\`ere equation, and under degenerations to a big class in the case of
Hessian equations.
