Hand-drawing a flat map to have the correct projection is difficult unless you're very experienced at it. What should be nice shapes in one projection start to stretch as you move it to another. In this case, you started with a very close relative of the sinusoidal projection, which radically distorts shapes in an attempt to keep areas correct. The first thing that I did was to reproject your input map into an Equirectangular projection, mostly because Equirectangular is the input to a lot of other software. I used one of those other pieces of software to produce the globe as well as a Sinusoidal image to compare to your original (it's a trifle off, but that's expected).
Original:
Original.png
Simple (Single) Sinusoidal->Equirectangular reprojection:
Plate-Carree.jpg
Equirectangular->Sinusoidal for comparison with original:
sinusoidal.png
As far as what you can do to reduce the distortion, you are probably going to have to redraw the map (or your chosen artist will need to). Another option is to redraw several regional maps in a shape-preserving (conformal projection) and assemble those into one final world image. It should be possible to do this by slicing your existing map and doing a simple skew before the final reprojection. The biggest difficulty in your case is that you have a lot of maps, so the process will need to be automated or you'll need to get one map and use that as a template to redraw the others. There will be some slight changes to your world along the way, but they shouldn't be too radical.
Getting the globe actually took longer than I like to admit because I was trying out a new process to simplify final assembly of frames generated by ProFantasy's Fractal Terrains product.