IMPORTANT EDIT 12-2015

There is this paper of Tokunaga "Irreducible Plane Curves of Albanese Dimension Two" which based on cited work of Kulikov constructs surfaces in P^2 with isolated singularities and Albanese dimension two.  In other words, the conjecture that for a smooth model of a normal in P^2 the Albanese dimension should be one is false.  This makes the answer to the question of the OP open. This information was given to me by R. Gurjar.

Original Answer- 
To the best of my knowledge this is a long standing open problem.  I cannot recall a reference, as this is something I studied in the 1980's, but I recall this being phrased as an unsolved problem from the 19th century Italian school.  The conjecture is that no normal surface in P^3 is birational to a smooth surface which has two dimensional image in it's Albanese.  One specific case of this that has been studied more extensively are Zariski surfaces:z^n = f(x,y) where f is a polynomial of degree n with only cusps and nodes as singularities. There are lots of information about when such a surface is irregular, but beyond that not much is known.  I believe that even if f is a sextic polynomial it is unknow whether or not the resulting surface can have 2 dimensional image in it's Albanese.  I have heard Catanese ask about the case where S is an abelian surface.