Wish I knew what to tell you about the axle... but I don't have the first clue unfortunately.
However - you have a Schwinn World Sport! That was my first "10-speed." I got mine when I was 14 or 15. Light blue paint, foam grips on the bars. Lost track of it post college. Thought ex-brother-in-law might have it stashed in a garage somewhere, alas it is gone.
Glad to hear someone else has appreciation for the model. Really hope the axle thing works out.