We systematically analyze a class of hexanomial functions over finite fields of characteristic $2$ proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over $\mathbb{F}_{q^2}$, where $q=2^n$, of the form \[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q}, \] we derive necessary conditions on the coefficients $A,B,C,D,E$ for APNness using algebraic number theory and algebraic-geometry methods over finite fields. Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates. We complement the theory with extensive computations: exhaustive searches over $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$, and random sampling over $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For $q\in\{2,4\}$, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family.