The clicking noise is the compressor overload kicking in and out due to over current draw because the compressor is not starting. If the start relay rattles when you shake it then it is gone and needs replacement. These are solid state devices with no moving parts. checking the windings is a good idea to confirm no opens or shorts occured because of the failure of the relay, but you have to remember that a mechanicaly stuck compressor will give good readings also. The winding readings should always add up, 3 pins, common to start reading + common to run reading = total reading start to run (with some slight variance)
You are deffinetly on the right track and a new relay should cure your problem.