Hmmm usually when it blinks fast means either a bulb is bad or possible bad connection between one of the turn signal lights n harness. Check em all out and yes check fuse & relay electrical systems are pretty simple.
When I purchased my 05 cummins it came with the issue as you have described "left" activated "right" and "right" did nothing or other way around... Solution was replacement of the Turn signal/wiper controllerMine has recently started actin up too, I click left and the right blinks then the left then nothin at all. Then then the right and the right blinks for a few then the left. I havent dug into it but I beleive the turn signal switch in the steering column is the culpret. Maybe this helps?
Keep us updated on the issue. Just to make sure, yur not having issues with turning left blinker on and the right coming on instead or other way around or just not doing anything. Its more of a turn right and blinks multiple times fast correct?