Which functional group is more polar in a drug: primary amide or a carboxylic acid?
It depends on your exact definition of polarity
The measured dipole moments of formamide (3.7 D)1 and formic acid (1.4 D)2 in the gas phase indicate that the amide is intrinsically more polar. 

However, this neglects additional factors relevant to the discussion: ionization and solvation. Under physiologic conditions, carboxylic acids are generally deprotonated resulting in a formal charge separation with a -1 charge delocalized across two oxygens. Depending on what parameters you are concerned with, this may be a more profound effect than the difference in gas phase polarity.