Usb-TTL serial module use true 3.3V uart.
You port setup was ok, because you menage to get signal from modem, to pic, and send it to pc from pic...
Also TX from pic is ok, because you get result(post 17) on PC when you connect USB module to modem tx...
So check your ground, and check pic input type, maybe 2.8V isn't enough to ensure logic 1.
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