Follow Chad's advice.
A slight modification, if I may, would be to disconnection the black wire off the horn relay before you start. Put your meter between this point and a good ground. This way, there is nothing else connected on the black wire. If I have to guess everything was ok before the turn signal switch was cleaned. Suspecting that somehow the black wire is pinched. Maybe leave your meter attached to the black and ground, and watch it has you take apart the upper column.
A slight modification, if I may, would be to disconnection the black wire off the horn relay before you start. Put your meter between this point and a good ground. This way, there is nothing else connected on the black wire. If I have to guess everything was ok before the turn signal switch was cleaned. Suspecting that somehow the black wire is pinched. Maybe leave your meter attached to the black and ground, and watch it has you take apart the upper column.