The silicon tubing method mentioned above works well.
There are dual dimension shaft couplers available, but they are not cheap.
If you have the tools and patience you can make your own with a piece of
scrap 9/16" rod about an 1" long. Just drill a 3/8" hole in one end, a
3/16" hole in the other and a couple of small set screw holes to secure
the coupler to the shafts. The trick is getting those shaft holes exactly on center.
Another option would be turn the end down on the 3/8" shaft to 3/16" and then just use a 3/16" coupler. This will lower the amount torque load that the shaft can handle, but I think the z-axis torque is low enough to be supported by only 3/16" steel.