In this paper, we study joint adaptive M-QAM modulation and power adaptation for a downlink two-user non-orthogonal multiple access (NOMA) network. Without sacrificing bit error rate (BER), joint adaptive transmission can fully utilize the time-varying nature of wireless channels, by allowing both power and rate to adapt to channel fading. Two adaptive power allocation strategies, namely, Scheme 1 and Scheme 2, each of which guarantees the minimum target rate for one user while supporting the highest possible rate for the other, are first proposed. Then, based on the two power schemes, the performance of joint adaptive transmission in terms of average spectral efficiency (SE) is studied for continuous-rate and discrete-rate modulation, while guaranteeing the minimum required rate and BER requirements. With the focus on practical discrete-rate M-QAM modulation, it is proved that for the strong user in Scheme 1 and the weak user in Scheme 2, their average SEs converge to the minimum target rates. In order to further increase the total transmission rate, we then propose a dynamic rate and power adaptation (DRPA) algorithm, aiming to increase the rate of one user without sacrificing the rate of the other. It is shown that at high SNRs, the DRPA algorithm allows the strong user in Scheme 1 and the weak user in Scheme 2 to continue to increase their transmission rates until reaching the highest modulation order that the system can support. Hence, the total transmission rate can be greatly increased at high SNRs due to the adoption of DRPA, by allowing both users in each scheme to reach the highest transmission rate in the system.