Bob, you may have more than one issue. First, I think you should know that flasher units that work fine with regular incandescent bulbs may not function at all with LED bulbs. Even the flashers marked "Electronic" or "Heavy Duty" relies on a certain amount of current flowing through the device for it to function. Special LED flasher units are available, but most have different physical pin-out spacing from the standard flasher units. You could add LOAD RESISTERS to allow the unit to flash, but that seems like a silly idea to me (but it can make the flasher work). LED lights take much less power (current flow) so you may have to deal with that.
You also should know that the HAZARDs and TURN SIGNALS are really two separate circuits sharing the same bulbs. The TURN SIGNALS are powered by the GREEN wire (switched hot with the ignition key) while the HAZARDs are powered by the PURPLE (always hot) wire. The hazard switch disables the function of the turn signals by breaking the circuit from the turn-signal flasher to the turn-signal switch, and at the same time energizes your generic relay to connect the left and right sides in series with the hazard flasher. (The PR or purple with red tracer wire is powered by connecting 1 and 2 of the hazard switch when the hazard switch is turned on -and disconnects 4 from 3.)
But, you say the relay clicks when you turn on the battery power (or screw down the post). That should not happen. Somehow, +12v is getting to the PR wire without the switch being turned on. My guess is your hazard switch is not wired properly, or is broken. You need to be careful with this diagnosis because you are working with some wires that are always powered. As a test, you may want to pull the fuse on the purple wire to see if the relay no longer clicks when you connect the battery. You may decide to solve one problem at a time -the HAZARDs or the TURN SIGNALs.
Good luck, I just went through this on a TR4A rewire, but the difference was in the way the new LED style flasher units need wired differently.