Like drone said front tire issues can be a cause of a low speed headshake. More air or less can make a difference...change just a little at a time to experiment.
Air in the sidecar tire can also make a difference. Usually more air over there will help take the headshake away.
Spokes on sidecar wheel should be checked.
Mounting system needs to be checked closely. If this headsahkle just began I would really want to go over that closely. I see in a previous thread ( http://www.advrider.com/forums/showthread.php?t=622528
) there were some concerns about the mounts. This was a while back and it seems as though you have run this machine a lot ....but... time can take it's toll on things. Any flex over time can be an issue in time. Heim ends should be checked closely for any play. Mounting points everywhere need to be checked etc etc,.
Anything that can create a movement within the assembly or a side to side movement should be ruled out. This includes steering head bearings, wheel bearings, wheels, swingarm pivot areas etc..
IF you have a poor mount and put on a damper and th eheadshake goes away that does not fix the poor mount. Bandaids will not heal an issue that needs to be addressed. Yes, a damper still may be a good thing regardless of what I am saying here but it is not a bad idea to rule out other issues especially if safety is invloved prior to slapping adamper on.
I don't want to sound like a jerk but am lost for the best way to say things sometimes. I do apologize for my poor way of explaining this.