Resolution of Barber paradox
No, you don't need to specially add any such clause.

Here's where I think you might have gone wrong.  The result is not 3 clauses (3 conjunctions).  The result is a bunch of clauses.  We get one clause of the form $\neg Shaves(Barber, x)\  \lor\ \neg Shaves(x, x)$ for each $x$, one of the form $Shaves(Barber, x)\  \lor\ Shaves(x, x)$ for each $x$, and one last clause $Shaves(Someone, Barber)$.  Thus, if there are $n$ people ($n$ possible values of $x$), the CNF form has $2n+1$ clauses in it.

Finally, note that the Barber is one of those $n$ people.  Thus, for the clauses corresponding to the case $x=Barber$, we get a clause $\neg Shaves(Barber, Barber)\  \lor\ \neg Shaves(Barber, Barber)$ and a clause $Shaves(Barber, Barber)\  \lor\ Shaves(Barber, Barber)$.  Now you should be able to find a contradiction in there.