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If the complex structure $J$ compatible with $\omega$ is integrable for $X$, then of course you have $\partial\bar{\partial}$ lemma. A natural generalization of the $\partial\bar{\partial}$ lemma in symplectic geometry is given by the "symplectic cohomology" introduced by Tseng and Yau: http://arxiv.org/pdf/0909.5418v2.pdf. Here you need to replace the operator $d^c$ by $d^\Lambda$, and the definition is analogous to that of the Bott-Chern cohomology. Just as the Bott-Chern cohomology measures how far the $\partial\bar{\partial}$ lemma is away from being true for a general complex manifold, the "symplectic cohomology" introduced by Tseng-Yau measures exactly the same thing for a general symplectic manifold. Of course the terminology here is a little ambiguous, because usually symplectic cohomology is defined by a direct limit of the Hamiltonian Floer cohomologies $HF^\ast(X,wH)$.