2001 F350, 160K, 4" cat back & intake mod.
Turn signals don't work/flash.
Hazzard's do work, except the front driver's side does not blink when hazzard's are on. All three of the others do flash. The right dash light flashes when hazzard's are on, not the left.
All bulbs are good. All fuses (inside & out) are good.

 

It could be (at least to me) the turn signal stalk going bad or the flasher unit itself going bad as far as which is quicker to replace ease wise hard to say which to start on first, but those two items are most likely to be the root cause of the issue.

 

The weirdest thing to me is the hazzards all flash but one!, and the relay makes that "clicking" noise when engaged. Yet there are no lights or clicking when turn signals are engaged.

 

As I stated the most likely issue is the switch/stalk or the flasher its self (I would start at the stalk).

 

"Well there's yur problem"!
Went and bought a new multi-function switch, took the column apart and when removing the old switch I noticed the longer molex connector had that locking tab broke off and so the connector was coming out! Soon as I pushed it back in, everything worked! I figured I would go ahead and put the new one in while I had it all apart and paid for anyway. Now I got a spare!
I put some Gorilla duct tape on the connector to help keep it from backing out again.

 

YAY \o/ glad you got it resolved