I don't understand, why a time offset in output waveform should be a problem. It's just, how the device works.
In the application note, there's only one VCO circuit, although it's not name as such: Figure 13, "Pulse Position Modulation". You'll find it with clearer waveforms in the 555 datasheet. But according to 555 properties, you can't achieve a constant duty cycle with variable frequency.You didn't tell about the intended frequency range and duty cycle, so I can't determine, if it may be acceptable to set the duty cycle to 50% for center frequency.