This paper studies the second-order coding rates for memoryless channels with a state sequence known non-causally at the encoder. In the case of finite alphabets, an achievability result is obtained using constant-composition random coding, and by using a small fraction of the block to transmit the empirical distribution of the state sequence. For error probabilities less than 0.5, it is shown that the second-order rate improves on an existing one based on independent and identically distributed random coding. In the Gaussian case (dirty paper coding) with an almost-sure power constraint, an achievability result is obtained using random coding over the surface of a sphere, and using a small fraction of the block to transmit a quantized description of the state power. It is shown that the second-order asymptotics are identical to the single-user Gaussian channel of the same input power without a state.