In order to address the signalling overhead and resource allocation problems of Machine-to-Machine (M2M) communications with non-orthogonal multiple access (NOMA), we propose a hybrid non-orthogonal random access and data transmission (NORA-DT) scheme. A novel design of NORA-DT protocol for M2M communications in cellular networks is firstly proposed. A power back-off scheme is introduced to adjust machine-type communications device (MTCD)'s target arrived power, and a closed-form analytic formula for the relation of MTCD's transmission power is derived. Based on the transmission power relation, the devices are clustered into a set of NOMA clusters. In the hybrid NORA-DT protocol, the cluster center MTCD transmits a extended preamble on behalf of the MTCDs in the same NOMA cluster on the physical random access channel (PRACH) for connection request. Base station (BS) can perfectly detect the preamble collisions in advance and schedules physical uplink shared channel (PUSCH) only to the NOMA clusters without collision. Then the MTCDs in the same NOMA clusters transmit data packets right after preamble transmission on the PUSCH to reduce the signalling overhead. By finding the optimal power allocation, we propose a low-complexity energy efficiency maximization problem for NORA-DT scheme. Due to the relation of MTCD's transmission power, we transform the problem into the function of cluster center MTCD's transmission power and solve it by difference of convex (DC) programming under the maximum transmission power constraints and minimum rate requirements at the MTCDs. A computationally efficient adaptive resource allocation scheme is finally proposed to improve the system throughput and resource usage. The optimal resource allocation between PRACH and PUSCH for any number of MTCDs can be learned by BS in advance, which avoids frequent computation. The analytic model is validated by simulation results. We demonstrate that the proposed NORA-DT scheme can significantly improve the system throughput, resource efficiency and energy efficiency performance.