Parasitic insensitive switched capacitor circuit
Some things to try:

To solve the offset problem try putting a large resistor to ground on the input of the VCVS, like 1e12 or larger. That way the simulation can find the DC operating point on startup. 

To solve the gain issue, use an actual integrator with resistors

Source:https://www.electronics-tutorials.ws/opamp/opamp_6.html