In this paper, we propose a novel joint source-channel coding (JSCC) approach for channel-adaptive digital semantic communications. In semantic communication systems with digital modulation and demodulation, robust design of JSCC encoder and decoder becomes challenging not only due to the unpredictable dynamics of channel conditions but also due to diverse modulation orders. To address this challenge, we first develop a new demodulation method which assesses the uncertainty of the demodulation output to improve the robustness of the digital semantic communication system. We then devise a robust training strategy which enhances the robustness and flexibility of the JSCC encoder and decoder against diverse channel conditions and modulation orders. To this end, we model the relationship between the encoder's output and decoder's input using binary symmetric erasure channels and then sample the parameters of these channels from diverse distributions. We also develop a channel-adaptive modulation technique for an inference phase, in order to reduce the communication latency while maintaining task performance. In this technique, we adaptively determine modulation orders for the latent variables based on channel conditions. Using simulations, we demonstrate the superior performance of the proposed JSCC approach for image classification, reconstruction, and retrieval tasks compared to existing JSCC approaches.