From your location, Villa Chalong is probably the best bet for a more gourmet supermarket along with a decent range of booze, but they are also pricey

If you can arrange a car and driver and want to do a single decent alcohol/groceries/mixers stock up one day early in your trip - consider a trip to the main Supercheap store in Rassada [just north of Phuket Town]
Apart from being a fun cultural excursion, they have the best prices on booze on the island, and a surprisingly decent range of both liquor and wines.
The prices at Supercheap easily beat any other chain supermarket on the island by 10-30% or more.

edit: keep in mind that there are restrictions on what time of day you can purchase alcohol in Phuket. 11.00am-2.00pm and 5.00pm-midnight on all days except certain gazetted Buddhist holidays (eg tomorrow there is a total ban on alcohol sales here for a significant Buddhist holiday)
Villa Market may be an exception to this rule on purchase times, a Phuket local should be able to confirm this or not