What happens to the gas inside cold drinks as the cold drink is frozen?
Well, you are certainly right in that the solubility of the gas will grow if you decrease the temperature, and so it will continue all the way down to the freezing point. That's when things get hairy.

Ice is quite different from liquid water. It does not dissolve any $\ce{CO2}$ to speak of.(*) So pretty much all $\ce{CO2}$ will leave the solution and go to the gas phase, adding its pressure to that of ice in an effort to tear the bottle open. (Some plastic bottles manage to withstand it, some don't.) The dissolved gas and whatever little salts are there will lower the freezing point of water and in fact make it an interval rather than a point, but that's another story.

After thawing, much of $\ce{CO2}$ will happily go back to the solution (though not instantly), and the drink will be more or less restored to its original state.



(*) For any physical quantity, there are layers upon layers of scientists from different fields, each confined to a certain range of the said quantity and calling everything above it "infinity" and everything below it "zero". The levels of dissolved $\ce{CO2}$ in the ice core samples from lake Vostok are quite a big deal for paleoclimatology. Not my field, though.