>>One part of the problem may be an improper command to close the connection and reset the modem in the end of the session - which may explain why it works once. The modem may have hung up, but doesn't necessarily mean it detected the success of the hung up, or that it returned to its regular state.
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>What happens if one is connected to a fax machine which they then use as a modem. Is that possible?

One thing I learned about modems (and I've installed about fifty of them, from 1200 to 56000 baud, plus the DSL now) is that anything is possible, and anything may take incredibly long to achieve.

One thing about a fax machine is that it's permanently in the "waiting for a call" state, while for PCAW it should be "I need to call", unless you're set as the called party, but I figure if you're running a session, you're calling, and the other party is waiting for a call. Also, fax has a different protocol, so I'm not quite sure what does it take to switch from "awaiting fax" mode into "initiating modem session" mode back into "awaiting fax" mode. That may lead into offering a human sacrifice in shape of any younger geek who will be willing to RTFM the whole command set of the fax&modem and to compare the docs with actual command sequences sent by PCAW.

In late DOS days, these things were getting standardized, but it all stopped by the end of the 90s, and I'm not sure that the development of modems really stopped. So your modem may need some command supplied differently, and your PCAW, being years older than the modem, may not know about it. But, AFAIK, every DOS communication package had to have a place where you could edit the command strings, you only need to know them - and that's up to the poor guy who has to learn them.

Make sure you tell the agency the exact model of your fax+modem. They may know the winning combination.