In this paper, we propose adaptively loaded set-partitioned offset quadrature amplitude modulation (SP-offset-QAM) orthogonal frequency division multiplexing (OFDM) for low-cost intensity-modulation direct-detection (IM/DD) communication systems. We compare this scheme with multi-band carrier-less amplitude phase modulation (CAP) and conventional OFDM, and demonstrate >40 Gbit/s transmission over 50-km single-mode fiber. It is shown that the use of SP-QAM formats, together with the adaptive loading algorithm specifically designed to this group of formats, results in significant performance improvement for all these three schemes. SP-offset-QAM OFDM exhibits greatly reduced complexity compared to SP-QAM based multi-band CAP, via parallelized implementation and minimized memory length for spectral shaping. On the other hand, this scheme shows better performance than SP-QAM based conventional OFDM at both back-to-back and after transmission. We also characterize the proposed scheme in terms of enhanced tolerance to fiber intra-channel nonlinearity and the potential to increase the communication security. The studies show that adaptive SP-offset-QAM OFDM is a promising IM/DD solution for medium- and long-reach optical access networks and data center connections.