Same problem on my '68 C-10. Turn signal canceling cam. Most likely the springs in there are shot or missing. It's also possible that you have a broken part in there that won't let it function correctly. It shouldn't be hard to pull it apart and tell. Obsolete Chevy should have the parts that you need. They are in Nashville, GA. I can get some other info of theirs if you need it. Hope this helps. John.

1968 Chevy C-10, 250 and 3 speed on floor, Hedman headers, Offy intake, Edelbrock 600cfm 4 barrel, HEI
1966 Chevy C-10, motor and tranny went into the '68. Waiting to be restored.