Four-dimensional computed tomography (4DCT) is essential for medical imaging applications like radiotherapy, which demand precise respiratory motion representation. Traditional methods for reconstructing 4DCT data suffer from artifacts and noise, especially in sparse-view, low-dose contexts. Motion-corrected (MC) reconstruction is a blind inverse problem that we propose to solve with a novel diffusion model (DM) framework that calibrates an adaptive unknown forward model for motion correction. Furthermore, we used a wavelet diffusion model (WDM) to address computational cost and memory usage. By leveraging the prior probability distribution function (PDF) from the DMs, we enhance the joint reconstruction and motion estimation (JRM) process, improving image quality and preserving resolution. Experiments on extended cardiac-torso (XCAT) phantom data demonstrate that our method outperforms existing techniques, yielding artifact-free, high-resolution reconstructions even under irregular breathing conditions. These results showcase the potential of combining DMs with motion correction to advance sparse-view 4DCT imaging.