Problem in DC offset with a summer op-amp
Your offset is added before the gain is applied- you can think of it as 1.0V referred to the input (input offset). 

If you want to provide output offset (for example, to center the signal in the range of a unipolar ADC) then you should add the offset after the gain is added. 

One way to do that is to leave the output amplifier gain constant, adjust the offset to compensate (halve it in your case, or use a 2K resistor in series with the offset rather than 1K and stay with 1V), and change the gain by adjusting the resistor shown below: 



Note that if you actually build this circuit you'll need a bipolar supply for at least the first op-amp since it otherwise won't be able to follow the input below ground, and it will typically have to be a bit larger (as much as a few volts or as little as some tens of mV depending on the op-amp) than the largest signal expected at the input. So if you want to handle a +/-10V signal with some op-amps, you might choose +/-15 supplies.