tunable Wave form generator design with op-amps Multisim
The frequency reduced when you added the zener clamps because the "aiming" voltage for the integrator reduced i.e. the voltages at the input to the integrator determines the rate at which the capacitor can charge or discharge - lower the voltage (by clamping it) and frequency naturally reduces.

For the 2nd scenario where you are trying to make the integrator charge at different rates depending on the input polarity, the average value of R2 and R5 should equal R2 in the previous circuit. Clearly this is not the case because in the original circuit R2 was 11k while the new circuit has R2 at 2k and R5 at 10k (average is 6k). Given that diodes won't be perfect (i.e. possess some volt drop when forward conducting), there will be other errors to consider.

Why not just try the following circuit - note the "symmetry" pot: -



Also, get rid of the 741 op-amp if you want decent performance at higher audio frequencies.