Ice is preferred if your cooler has a drain that can remain open; oysters -unlike clams- are fine to be in direct contact with ice. The important thing is to prevent these salt water gems -oysters and clams alike- from becoming submerged in fresh water. This will drastically change their flavor and can make them unsafe for consumption.
Oysters can actually remain alive for up to 10-12 days (this is not recommend of course) when kept buried in ice and allowed to constantly drain.
Or to keep it simple, remove from ice, keep refrigerated and covered with a wet towel for up to 8 days.
Properly harvested, handled, and stored, fresh oysters will keep better than you might think. So no need to eat them all at once. Above all else, keeping shellfish COLD is key! If having your oysters raw, the colder the better. Enjoy!