A pair of sequences is said to be an almost-complementary pair (ACP) if they have zero aperiodic autocorrelation sums except at only one position over all the positive/negative time-shifts. Having correlation property very close to that of Golay complementary pairs (GCPs), ACPs may be used as an alternative to GCPs in many applications in communications and radar. For high-rate code-keying OFDM communication, we construct novel 16-QAM ACPs from three new classes of quadratic offsets, leading to three large sets of 16-QAM almost-complementary sequences with maximum peak-to-mean envelope power ratio (PMEPR) of 2.4.