LM358: Output clamped at 9.63V with 3.5 noninverting gain.

Hi Adam, 

Welcome to E2E! :) 

LM358 is not able to swing to the positive supply rail (rail-to-rail output), so you may be seeing some output swing limitation. In our Electrical Characteristics table in the datasheet, we have a specification for output swing, showing that the LM358 can only swing max 3V from the positive rail (Vs =30V, RL > 10kohm). 

If you need a device that can swing closer to the positive rail with the 12V supply, I recommend using the OPA2990.

Please let me know if you have further questions.
Thank you!

Best Regards,
Ashley

Section 3.3 of the LM358 app note explains how the output drops with load current; see below. You should use higher-valued feedback resistors.