1) If you induce a sudden chance in pressure you will indeed take the system out of equilibrium. Lets consider a closed recipient containing some liquid and some gas above. If you suddenly inject more gas into the recipient, the gas pressure is initially going to raise fast and then going to decrease slowly while part of the gas is dissolving, until the system reaches equilibrium again. However, this new equilibrium will have a higher gas pressure AND greater amount of gas dissolved in the liquid when compared to the initial equilibrium. Of the amount of gas you injected part will be dissolved and part will stay above the liquid, contributing to the higher gas pressure. It makes no sense to have pressure under the initial value, this would only be possible if the system hadn't been at equilibrium at the start.
2) Shaking doesn't increase the pressure, unless you shake the bottle long enough to cause a considerable increase in the temperature inside, but that would be quite an impressive amount of shaking. If you could take the system out of equilibrium by releasing some of the dissolved gas from the liquid phase, then that would increase the pressure, but shaking doesn't accomplish that, and I believe it wouldn't be possible to shift a system out of equilibrium like that without transference of matter or heat and without a change in volume or temperature.
There is a very interesting video by Veritasium on youtube about this bottle issue, I recommend you to watch it: