Replace the TWC Modem with an aftermarket one.
TWC Telephony modem and POTS are not the same. Just because it worked on a POTS's line doesn't mean it will work through that modem.
You might get it to to work sometimes at whatever baud rate you are setting it at, but if they say it only works at 9600 baud they could just mean it only works reliability at 9600 baud and nothing higher