Easy. Your brake lights feed through the turn signal. That's how it turns off one brake light and flashes when that side turn is selected.
So the problem is..........
either the brake light switch is not getting power
or the brake light switch is bad,
or the signal switch is bad
So get your light/ meter and probe the switch, up under the dash near the brake pedal. Probe the connections, one should be hot and the other cold. When you depress the pedal, BOTH should be hot. Then move to the turn signal connector "just to be sure." Check that you have brake light power going into the signal switch with pedal depressed
If you do, it's the column switch
If you get power to the brake switch but nothing out, it's a bad brake switch.
You have a shop manual?
This is the closest on this site, does not cover the dash/ cluster differences for Barracuda, but the rest of the car is substantially the same, although '69. Go thank AbodyJoe:
[ame]http://www.abodyjoe.com/pictures/Misc.%20car%20info/69%20dodge%20service%20manual.pdf[/ame]
Here are somewhat useful but not always complete diagrams from MyMopar
http://www.mymopar.com/index.php?pid=31
Two diagrams:
http://www.mymopar.com/downloads/1968/68BarracudaA.jpg
http://www.mymopar.com/downloads/1968/68BarracudaB.jpg
On diagram "B" above find "stoplight switch" along the left side of the page. Note that a pink goes from the emergency flasher to one side of the stoplight switch. This is a junction point and feeds power to the switch.
When the switch is made, power goes out on the white off the left of the page to diagram "A" and goes over to pin "6" of the turn signal switch connector. Note that there are TWO white wires, but only one goes to a "corner" terminal, which is the one you want to check
If you have power there with pedal down, and no stoplights, it's a bad signal switch.