Electric field in coaxial cavity
I would look at this as two (strongly) coupled cavities. One cavity with a center conductor and one with no center conductor. 

Then you can express the field in the lower section as a sum of modes with the appropriate boundary conditions (longitudinal field going to 0 at \$r=r_r\$), and the field in the upper section as a sum of different modes (with no interior boundary). You'd then have a boundary condition at the interface between the two "cavities" where the total field must be equal for the two solutions at \$z=l_d\$.

I don't think you will find an analytical solution. For one reason, because we know the coaxial structure will have TEM modes, while the simple cylindrical structure won't have TEM modes. Therefore the modes of the coupled structure will be complicated combinations of the modes of the component structures.