"Right now, I've got another relay on order, hoping that may help."

Did you decide you had the problem equally on right and left? I thought I made it rather clear in my original reply, the relay would not know how to discriminate.

I've had the same problem our friend with the camera has documented so well, so I believed his posts would have led you to a closer look and a fix. Before it looks as bad as his, the problem is quite hard to see. A film of oxide may be invisible to you but high resistance to electrons long before it turns into a burned mess.

If you really want to get to the bottom of this, after you swap in your new relay and it does the same thing, make sure the 4-way hazards cause the same behavior. Then, put your multimeter in the configuration to measure current, and do this by substituting it for fuse 13. Compare left and right in turn signal mode. Being as fickle as it is, you might find the problem exists both left and right, but your car feels more politically correct when making a right turn. Sadly, most people will swap the part they believed would fix it and regardless of what really fixed it will give the credit to the money spent.
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Art Benstein near Baltimore

A boiled egg in the morning is hard to beat.