unistudent,
your answer to the problem was correct. TEA gets protonated.
Concerning positive and negative ions, don't forget that you are in an aqueous solution.
So the ions will stick together and form electrically neutral molecules only if such neutral molecules are energetically more stable than the dissociated (and solvated) + and - ions.
For triethylamine hydrochloride, the solvated ions are more stable.
In other cases the bond between the ions is so strong that it prevails over the solvation energy. But that's generally more frequent for inorganic species. Example: AgI, BaSO4, HgS...