As far I know carboxylic acids have a OH group but during dissociation only hydrogen gets separated. The CO moiety is polar, demonstrating that oxygen is much more electronegative than carbon. Why does the OH group not get separated from carbon and behave like a base?
Well it can act as a base depending upon what kind of other substances are there in your mixture. As a matter of fact any substance can act as a base or an acid depending on the substances that are present. Also there are more ways of defining a base (here I think it is a good idea to use the Lowry-Bronsted definition in which a base is anything that can capture H+ and the OH of a carboxylic acid can indeed capture a H+ that may come from a strong acid like H2SO4.And indeed as mentioned before it leaves as water not as HO- the reason why it doesn't act as a normal base (for example NaOH where you have HO- in the solution when you dissolve it in water) is because that one is an ionic compound Na+ and HO- while in a carboxylic acid the bond between C-O is a covalent bond even if it is polarized so it needs some other conditions for its departure to be reasonable (energetically speaking) thus a HO- can not just leave the compound.
Well. Carboxylic acids are acids in water. In other solutions, they may act as a base. For example, in the esterification process, they are mixed with an alcohol in the presence of sulfuric acid (or any other dehydrating agent). It has been possible to show, by using unusual isotopes of Oxygen, that the oxygen bound to H in the -COOH group is eliminated, as if it was a base. The point is that the Oxygen atom used for the -C-O-C- bridge, typical of esters, is coming from the alcohol and not from the carboxylic acid.