With increase in pressure in carbonated drinks, dissolution from CO2(g) to CO2(aq) does increase. With increase in the aqueous carbon dioxide, the volume increases too as the equilibrium is shifted to the right. Refer to the Le Chatelier's Principle again.

The reactions involved-

dissolution : CO2(g)-->CO2(aq)

hydrolysis : CO2(aq) + H2O(l) --> H2CO3(aq)

ionization : H2CO3(aq) --> HCO3-(aq) + H+(aq)