First, I assumed you meant the right side signals stay dark when activated? I ask because a dead filament in a turn signal side of a bulb will keep it light up the bulbs but they will not blink.

I have not seen a diagnostic for the switch other than pulling the steering wheel and examining it. For example, mine broke twice, once I found a broken spring for one side and replaced it. The second time, I diagnosed it from a strange set of rear lights issues. That time I replaced the whole switch.

However, (thinking only) using the wiring diagram to find the +12 input and right and left output, you might be able to unplug the switch from under the dash by the column and, using an ohmmeter, check the continuity on both switch sides. If the left side circuit closes when activated and the right does not, it's in the switch.

There are many discussion in this forum about that part, you may want to research those first.

Note that there are different switches for various different '68 Firebird options.

Attached are the two pages from the '68 Diagnosis Manual on the directional signals subject.

They don't do much testing, mostly just replacing parts.