Differential Amplifier with Large Capacitive Load
Where is the supply decoupling?

I find R1,2,3 rather low, are you sure the INA149 can drive into a 1k load satisfactorily, try using 10k here?

The TLE2426 usually likes quite a lot of cap on the output, I have seen these act out if this is missing, try 100uF from its output to ground. 

That right hand opamp looks like it is trying to be a MFB filter but is actually wired with its inverting and non inverting inputs reversed, that is not going to work. 

What does your ground layout look like? when you close those switches the '149s are having to source quite some audio current, and if that disturbs the ground for the TLE2426.