Yeah, CTS and RTS is useful for the module since you may overflow the module buffer (instead of the module overflowing your UART buffer). With proper HW flow control, hopefully your device UART respects the signal and pauses the tx until it is clear again without you having to code the pause yourselves. It can happen when the GSM bandwidth is lower than the UART bandwidth.

The module suddenly talking should also be handled by your device UART gracefully. When your rx buffer is full for whatever reason (not reading it for a long time?), the module won’t be sending anymore data until you read your rx buffer. Theoretically, no data should be lost that way.