From the electrical diagram for my 63...
The harness connector on the turn signal switch on the lower steering column has slot connectors in a 3-2-1 row configuration.
On a 63 the 2-connector row feeds the rear turn signals, with a purple wire going to the driver side tail light and a violet (ie, light purple) going to the passenger side tail light.
I think the 1-connector "row" is a 12v wire that is turned on by the brake light switch to feed both the purple and violet wire connectors via the turn signal switch.
The outer connectors in the 3-connector row go to the front turn signals and to the dash turn signal indicator lights, so the dash indicator lights probably look normal despite malfunctioning rear turn signal lights; the center connector in that row is the pulsed 12v power feed coming from the turn signal blinker module at the fusebox, I think.
If you have power to both the purple and violet wires at the turn signal switch with the brake on, but with the brake lights off you see pulsing to the purple wire with the left turn signal on, but not to the violet wire with the right turn signal on, then the problem is likely in the turn signal (since the brake lights are both working).
There is a slider on the side of the turn signal switch that is moved by a stiff-wire cable (from the turn signal lever) that passes through 2 tabs, one on either side of the slider; the tabs "trap" the cable/wire by exerting pressure on it. The slider rides in a cut-out slot in the turn signal switch metal housing; when the turn signal lever is moved up, the slider moves to one end of the slot; when the lever is moved down, the slider moves to the opposite end of the slot. With the turn signal lever in the off (center) position, the slider should be centered on/in the slot. So begin by checking that slider for correct positioning while moving the turn signal lever to signal left and right turns, and adjust the cable wire in the tabs if needed. If the slider does seem to be positioning correctly in the slot in both directions, but the violet wire connector on the switch is not getting a pulsing 12v signal when the lever is in the right turn position (yet the right brake light is working) then it's likely the switch is the culprit.
You probably could test this also (without power): disconnect the harness from the turn signal switch, and use a multimeter to check continuity (ohms scale) between the center connector of the 3-connector row and each of the two connectors in the 2-connector row, with the turn signal lever in each of it's positions... I think you should have continuity to only one (or the other) of the two connectors with the slider at either end of it's travel.