No. Oxygen molecules would not "sink" because they possess thermal energy.
The thermal energy possessed by an oxygen atom
= (3/2).k.T
where k is the Boltzman's constant (=1.38 x 10^-23 J/K)
T is the temperature in Kelvin scale
thus, thermal energy of oxygen atom at 300 K
= (3/2).(1.38x10^-23).(300) J = 6.21 x 10^-21 J
Taking the mass of an oxygen atom to be about 2.7 x 10^-26 kg (i.e. a weight of 2.7 x 10^-25 N) , such thermal energy is sufficient for an atom to "fly" to a height of 6.21x10^-21/2.7x10^-25 m = 23 000 m or 23 km above the earth surface.
Since oxygen is heavier than air, if the oxygen is trapped inside a balloon, it will sink. So as a helium gas balloon will rise. (Helium is the lightiest gas.) But if oxgygen is left free in the environment, it will not sink. It will move freely in the environment until it is evenly spaced within the environment.