How do I compensate for Exponential decay in an op-amp circuit
The circuit is a square-wave oscillator followed by a Sallen-Key LPF tuned to attenuate harmonics above the fundamental. Since the 3rd harmonic of a square wave is 1/3 of the fundamental, the second-order Sallen-Key filter is not going to to do a great job, but it should look more-or-less like a sine wave at the output (once it has been operating for a few cycles). 

If you are seeing something else, re-calculate your values and make sure the parts you are using match the theoretical values closely enough. 

The oscillator would be about as bad (actually a bit more stable) if you used the 555 followed by the low-pass filter (LPF).