We experimentally investigate transmitting high-order quadrature amplitude modulation (QAM) signals with carrierless and intensity-only measurements with phase retrieval (PR) receiving techniques. The intensity errors during measurement, including noise and distortions, are found to be a limiting factor for the precise convergence of the PR algorithm. To improve the PR reconstruction accuracy, we propose a distortion-aware PR scheme comprising both training and reconstruction stages. By estimating and emulating the distortion caused by various channel impairments, the proposed scheme enables enhanced agreement between the estimated and measured amplitudes throughout the PR iteration, thus resulting in improved reconstruction performance to support high-order QAM transmission. With the aid of proposed techniques, we experimentally demonstrate 50-GBaud 16QAM and 32QAM signals transmitting through a standard single-mode optical fiber (SSMF) span of 40 and 80 km, and achieve bit error rates (BERs) below the 6.25% hard decision (HD)-forward error correction (FEC) and 25% soft decision (SD)-FEC thresholds for the two modulation formats, respectively. By tuning the pilot symbol ratio and applying concatenated coding, we also demonstrate that a post-FEC data rate of up to 140 Gb/s can be achieved for both distances at an optimal pilot symbol ratio of 20%.