Seabee,
Which turn signal bulbs do you have installed in front and rear, 1157 or 1034? Many years ago when the 1157 came out we would occasionally get someone come in with a rapid flash on one side. Sometimes it would be due to a heavy duty flasher with a 1034 bulb installed in lieu of an 1157. (or was it the other way around...Darned...Getting old ain't no darned fun. Keep forgetting things that was well engrained back then). Other times a mis-keyed bulb installation where it went in backward and twisted with little effort due to poor bulb manufacture or weak/worn aluminum socket. I would check for the proper bulbs and installation and maybe consider replacement of both front and rear bulbs just for grins and maybe a flasher unit too. They are cheap enough. I would start with a close review/inspection of the system components first.
I am most curious on the Rt turn signal operation regarding both sides' tail/turn lights function with the headlights OFF, Rt Turn signals ON and brake lights ON, then with headlights ON.
In addition to post #18 suggestions here are some more detailed tests to consider:
In the dark of night turn on the right turn signal. Walk around the truck observing
all external lights for any indication of illumination, dim or bright. Are side marker lights working properly? Try the same thing for the left turn signal. then do both again with the parking lights then headlights on. Looking for errant electrons wandering in the wayward paths of lesser resistance.
Disconnect the Brite Box and, if possible, connect to original configuration to eliminate the Brite Box and check the operation of the turn signals. Re connect if no change.
Next:
Headlights off.
Remove Rt front turn signal bulb.
Observe filaments' physical conditions.
Confirm all bulb continuities with an ohmmeter.
Activate Rt turn signal.
Observe light actions. Are they still rapidly flashing, or not flashing?
If flashing measure voltages at turn signal and parking light contacts to bulb socket ground.
Have someone apply the brake to energize the brake lights, note light actions front & rear, and measure the contacts' voltages again.
Release brakes.
Turn on headlights and measure all contact voltages again.
Turn headlights OFF.
Turn right turn signal OFF.
Check bulb socket case to chassis ground for continuity as well as ground contact points for tightness and corrosion.
Re-install front TS bulb.
Repeat the above steps with the Rt rear bulb removed and measure Rt rear turn signal and parking light contact voltages without and with application of brake lights and headlights, also observing front lights' filaments for dim glow.
NOTE: When viewing any bulb activity it should be a naked bulb and shielded from as much light as possible to observe both elements for even the dimmest light activity. Include observing license plate and side marker lights. You may see spots for a while attempting to see any dim illumination of the parking/tail light filament with the bright element glaring away.



I would even remove both front and rear left side turn signal/parking lights from their lenses and observe the elements for very, very dim illumination.
Checking continuity from socket pins to ground would be a questionable test due to all the lamps in the circuits. Although you could compare readings and if something is significantly out to lunch in the readings, devote more time in that area. Removing them would be time consuming and possibly cause unintended boo-boos. Best used when all else fails.
If any observed voltage is low with the above tests, disconnect the light housing/socket from the harnesses and measure voltages directly at the truck-side plug contacts to chassis ground.
Open the 7 way tow electrical connector
Check for contact and connection conditions.
If it was installed using scotch-locks, check for splice and wire condition.
If factory installed, check harness plug condition.
On EDIT: When performing the voltage tests above be sure to include the 7 way tow electrical connector for voltage checks.
Maybe try hooking up a trailer to confirm operations before testing. This would add additional load and ground circuits to aide in TS. Disconnect prior to performing above tests.
I don't believe the turn signal indicator bulb would cause this issue. 193, 194, 168, 151(?) should all work.
Sorry for all the work suggested. Just trying to help narrow the issue down.
Best of luck.