The way to store a small quantity of dry ice for a week without spending a lot of money on a freezer that will actually keep it below sublimation temperature, or having a cylinder of the pressurized liquid and a "dry ice maker attachment" for the cylinder to make some when you need some, is to buy a larger quantity of dry ice and store it in a well-insulated container. A Dewar flask would be the best option; a large amount of styrofoam is less ideal, but can be good enough. Nested foam coolers (ie, a foam cooler with dry ice inside another foam cooler) can hold it for a while, but details will depend on variable factors. You do need to consider the ventilation of the area where it is stored, as it is possible to fill an area with the heavier-than-air gas and manage to kill yourself, or others, or pets.
As for the practicality of using it in cooking, marginal at best (and dry ice, in particular, may cause taste effects from dissolved CO2 in the cooled product (ie, sparkling .vs. still water.)
Your martini, for instance, could be much more easily super-cooled by storing the gin or vodka in the freezer - I suspect the vermouth might freeze (not a martini fan, but most wines will freeze in a common residential freezer, while most distilled spirits won't.) Perhaps you could make vermouth cubes.
Chris H's answer has good details regarding latent heat and use of ice/salt mixtures (I upvoted)
In commercial kitchen environments large quantities of liquids that require rapid chilling may be stirred with a "cooling paddle" - the "secret" to which is generally a water-based (or simply water) filling that is pre-frozen to make ice, and a large surface area to rapidly melt the ice, pulling the latent heat of melting from the hot liquid, without mixing in water.
Pouring it into a hotel pan in a shallow layer, and setting the hotel pan in another hotel pan full of ice is another effective approach.