Okay fellas, I now have a functioning 4 speed Esquire bike. Thank you very much, piercer_99. Since you told me there should be no problem using a grip shifter, I pulled the one off an old Raleigh that a neighbor donated to me a couple of years ago. I stripped the cable housing off the Esquire as it runs full length and used it on the cable off the Raleigh. I lubed the cable and housing with WD-40 and 3 in One oil and slid the old cable through it with no problem! Re-assembled everything and rode off. Much to my amazement, all of the gears work flawlessly now. You guys probably won't think this bike is much to look at, but I just love riding it. I think maybe it's the seat and handle bar positioning for me. Here's some pictures of the cursed shifter and the internal parts. I won't go into detail about how many time I put it back together and took it apart, because I lost count! I tried several different variations, and NONE of them worked once I put it back into place and aligned the red marks on the hub while keeping it in 4th gear. So, you can now see my Esquire with the grip shifter on it. Now I gotta find something else to keep me occupied and frustrated. Neighbor never brought the moped over. I may try to frankenstein a bike just for the heck of it. I'll let you know. Thanks again to everyone who offered help and opinions.
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