I was just in this boat months ago....sigh
Okay.. check to make sure the wires from the actual turn signal stalk/switch are not corrodid and falling out. The last repair guy to work on my car replaced this turn switch behind the steering wheel.
Now when a light bulb in the brake system starts to go out the blinker will move faster because it needs this bulb to add enough resistance to blink slow. This is also a problem for many people that want to mod their tail lights with LEDS in that there is not enough resistance so they end up buying a repeater to fix this problem.
So check the light bulbs in the brakes first. Make sure the leds are not melted down flat. If so try to reshape them, stick in some metal or just buy new bulbs and be do with it. As you turn those bulbs get loose and they blink fast.
Now as for the click noise that lets you know the signal stalk is locked. It should turn off after the completed turn. Well, this would be the actual turn signal switch. It's all on one unit that you must remove and replace. I don't recall how much this costed me? Even the new turn signal switch was acting a little funky and rather than replacing that one I just wedged it with some rubber plastic to make the wheel catch and turn off as it would otherwise just keep blinking.. long story.
Now you see where I'm pointing to those brake wires? You can try and wiggle those wires around. The black thing that the wires are going into also plugs into the white plastic turn signal switch.