All 4S models use the same replacement screen. In other words, there is no such thing as a "CDMA iPhone 4S Screen" or a "GSM iPhone 4S Screen".
To be completely clear, the iPhone 4 and 4S actually use the exact same replacement screen assembly. The only difference is in the plastic frame/bezel that the replacement screen is installed into. The GSM iPhone 4 uses one frame, the CDMA iPhone 4 a different one, and then all models of the iPhone 4S use yet another different frame.
So the moral of the story is that it's not a compatibility question. Perhaps the connector for the LCD on either the replacement screen or the logic board was damaged during installation. Or, since you said the phone was "pretty messed up", perhaps the logic board was already damaged when you got the phone.
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