Xi Chen has a theorem that says that rational curves on K3's in a linear system of dimension $>3$ are nodal. I suppose you don't need this curve to be rational, but his techniques might help you in your quest. At least his theorem tells you that you cannot expect too many cusps.
I couldn't find the paper online, here is the MathSciNet review:
MR1675158 (2000d:14057) Chen, Xi(1-UCLA) Rational curves on K3 surfaces. J. Algebraic Geom. 8 (1999), no. 2, 245–278. 14N10 (14J28)
The paper is essentially divided into two parts. The first part proves existence of rational curves in the linear system |OS(d)| on a general K3 surface S in Pn (n≥3 and d>0). The rest of the paper is devoted to the following conjecture: For n>3, all rational curves in the linear system |OS(1)| on a general K3 surface are nodal. The conjecture is proven in the cases n≤9 and n=11 and hence justifies Yau-Zaslow's beautiful counting formula, ∑∞g=1n(g)qg=q/Δ(q) for g≤9 and g=11. A basic ingredient in the proof is the degeneration of the K3 surface to a trigonal K3 surface, that is, a K3 surface with Picard lattice congruent to (2n−2330). The moduli space of trigonal K3 surfaces consists of countably many irreducible components (of dimension 18), and the author considers three of these in order to prove the conjecture in the cases described above. Various reformulations and generalizations of the conjecture are also introduced.