Marine grease or lucas red-n-tacky. Both are pretty sticky compared to typical axle grease, so they adhere vs getting all flung off. If I didn't have a tub of red-n-tacky, I'd get a tub of marine grease as it seems to hold up better...
Hard to see that second photo, very blurry. Since you got it out, now you can take it apart and see what's in it and that the pawl moves when the set screw/spring/ball bearing are removed.
Clean all the grease out, put a little on the metal to metal gears and leave the rest dry.
Tighten until the spring is fully compressed, then back it off 1/4 turn.
The pawl will allow it to run fine in first gear. But if it's stuck, it just won't shift no matter what you do.
Check your slipper, slipper pad and clutch in the clutch bell. If your slipper is slipping bad, when it does engage 2nd gear, the slipper just slips vs the truck going faster and the motor dropping in RPM. Once it slips, it glazes the pad, then it slips worse and worse until it gets so hot it...