Why Does Op-Amp Output Voltage Always Saturate (in a Non-Inverting Amplifier Circuit)?
Common op amps generally will not perform well near their rails - ground and 12 V in your case. Powering the amp with a 7 volt or higher +/- supply (14 volt total) would likely cure the problem.

Also note that you should set the unused op amp in the package to a gain of one to prevent oscillations that can disrupt your circuit.