If I disconnect the phone line out of the modem, the dail tone instantly
returns.

This morning when it happened, I checked using the Control
Panel/Modem/Diagnotics/Info and in AT14 was the "last dailed # T9". As
my ISP's is 7358647, I don't know what the modem or another program was
trying to do.

When I checked after a successful online session, the AT14 was "last
dailed # T7358647" as it should be.

In any event, it is not uncommon for modem to become damaged - esp during thunderstorms - such that it works except stays off-hook all the time, even when not in use.

You can examine modemlog to make sure no other app (like fax/ans machine software) is accessing modem to cause your problem
www.808news.com/56k/modemlog.htm

Also see
www.808news.com/56k/hangup.htm