Many LED bulbs are sensitive to polarity, but on a 67/68, the ground is the shell chasing. The ground lug for the dash/underdash is located a couple of inches to the right of the dash cluster harness break-out: look for a single black wire near where the turn signal wires break out (orange, blue wires). Brake and all turn signal lamps don't use this particular ground path; sounds more like a bad turn signal switch.
Randy, aren't the turn signal sockets hard grounded with an additional wire to the socket? They have the larger black sockets unlike the metal ones for the dash lights, which do ground to the cluster.
I agree to check the main harness ground behind the cluster. Also make sure there is a star washer used. The order on the screw should be wire loop first, then star washer last. You want the star washer in direct contact with ground surface. Some folks do this wrong and put it on the other side.
Also, it is very uncommon for owners/shops to miss installing the star washers at all the ground points at the engine and engine bay. The rear engine ground to firewall should have one at both ends, the headlight harness will be grounded in a couple spots with them and the negative battery terminal to block connection will have star washer too.