A method for making dynamic borehole azimuth measurements while drilling includes processing cross-axial magnetic field measurements in combination with accelerometer measurements to compute the dynamic borehole azimuth. In one or more embodiments, the cross-axial magnetic field measurements and the accelerometer measurements may be used to compute the magnitude of a cross-axial magnetic field component, a toolface offset, and a borehole inclination, which may in turn be used to compute the dynamic borehole azimuth. The disclosed methods may utilize near-bit sensor measurements obtained while drilling, thereby enabling a near-bit dynamic borehole azimuth to be computed while drilling.