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If the extension phone continues to get a dial tone, dial out, make calls, and receive calls, then your phone system and your cables are obviously in good shape.
The faxing electronics in your Canon machine - not so much. It's probably 'broken', simply.
One thing to try is, to check that the phone line goes into the 'L' or Line jack on the Canon, since the Canon won't work as a fax if it is connected some other way.
After checking on that, though, I would say that, after unplugging the Canon machine for at least 1 hour to rest and reset, if it still does not work, and if there has been no change in your phone service (esp. from 'landline' to 'digital' phone service, which often has problems with the use of fax machines) then you would have to conclude that your Canon machine's faxing electronics have been blown out (possibly by a surge on the line, e.g. from a local thunderstorm) or have 'broken' otherwise.
I'm sorry to have to come to this conclusion, but it seems inevitable.