If Harding lived, I could see him as the first president to resign, thus:
• He fires Denby, Daugherty, et. al. for betraying him and the trust he gave (indeed, IOTL, he came very close to physical violence with Daugherty). Reason: they cannot function until their names are cleared, and should thus devote their energies solely to clearing the matter
• Harding calls a press conference, saying:
+ he is aware of the rumors of scandals in his administration;
+ he does not know the degree to which the rumors are true;
+ in any event, as chief executive, he has the ultimate responsibility and is therefore stepping down because he cannot govern effectively until such time as the matter is settled
Coolidge succeeds Harding and runs for a term in his own right in 1924. Meanwhile, Harding returns to Ohio and is ultimately exonerated of any involvement, while the aforementioned are tried, found guilty, and serve time in a federal penitentiary. Florence Harding, disgusted that her husband resigned rather than fight it out, files for divorce. An uncontested divorce is granted in 1925. Harding himself enters a new career as a baseball executive with the Cincinnati Reds and is considered as a future league president until his death from coronary artery disease in 1929.