Hello Anita. Storing fresh melon properly can make a noticeable difference in both its crisp texture and sweetness over several days.
Between the two options you mentioned, both are viable, but slicing and placing the melon in airtight containers is generally the superior choice for your situation. Watermelon flesh is over ninety percent water and readily absorbs ambient odors from nearby food in your refrigerator. Airtight containers create a reliable barrier against odors while allowing you to stack the fruit neatly, which will immediately free up the shelf space you need.
When cubing or slicing the melon, ensure your cutting board and knife are completely clean to avoid introducing bacteria that accelerate spoiling. As the fruit sits, it naturally releases juice at the bottom of the container; pouring off this excess liquid after a couple of days helps keep the remaining pieces firm rather than soggy. Stored this way, it should maintain great flavor and texture for three to five days.
If you ever decide to keep a half intact in the future, wrapping the cut surface tightly with plastic film all the way around the rind edge works well too, but it does consume far more physical space and leaves the fruit more susceptible to drying along the cut plane.