Welcome to the Forum:
Has the Q500 ever flown successfully on the carbon fiber frame and using your controller?
Some of the description sounds sort of like you're already bound and just have a bad motor/ESC. Either way, we need to do some troubleshooting to determine the problem.
There are two diagnostics you can use to get an idea of the issue:
Q500 GUI :
Connect the Q500 to the Q500 Gui on your PC using the dongle that should have come with the original kit. The GUI will tell you if any sensors are preventing a bind. There is also a motor test routine to check the motors/ESCs. Let us know if you are not familiar with setup and use of the GUI.
Use the Binding Verification Document:
The guidance was written for an ST10 but will work the same for the ST12. The verification is not just "binding instructions". It is a verification of the MANY hidden clues in the binding process. The purpose is to identify abnormal conditions that are preventing a successful bind and to identify the exact point the binding process is failing. That information can assist in determining the best course to remedy the problem.
You will need to perform every step up to the point the verification does not match what you see. Do not skip any steps. Every step must be performed. Even the steps you have no doubt are already in the correct condition. After some preparation steps, the verification falls into a routine of directing an action, then verifying the correct response to the action. When you reach a verification step that does not match what you see, or if you are unsure if it matches, STOP. Report the step number back to the forum, and what you actually saw at that step. We can then go from there.
Q500 to ST10 Binding Verification