I wouldn't put it past Welles, but the reality was that Welles was busy doing the Mercury Theater's Sunday matinee show, and was arriving by taxi at the studio, minutes before "The Shadow" went on the air.
There's an audio show, "The Story of the Shadow", running about 2hrs, iirc, that goes into a lot of the anecdotes, including the "dropping the script".
The longest, and final Margo Lane was a Canadian actress that had a story of being at some ritzy cocktail party, and describing her radio roles to some people, boring the heck out of them, until she said "...and I do 'Margo Lane' on 'The Shadow'", which got the entire room's attention.
I do Hebrew calligraphy, which requires my eyes to be on the parchment I’m writing on, so radio shows, especially Rush, are the only entertainment I can have. Weekends, there’s no Rush, so I look for old-time radio shows.