An integrate-and-fire time-encoding machine (IF-TEM) is an effective asynchronous sampler that translates amplitude information into non-uniform time sequences. In this work, we propose a novel Adaptive IF-TEM (AIF-TEM) approach. This design dynamically adjusts the TEM's sensitivity to changes in the input signal's amplitude and frequency in real-time. We provide a comprehensive analysis of AIF-TEM's oversampling and distortion properties. By the adaptive adjustments, AIF-TEM as we show can achieve significant performance improvements in terms of sampling rate-distortion in a practical finite regime. We demonstrate empirically that in the scenarios tested AIF-TEM outperforms classical IF-TEM and traditional Nyquist (i.e., periodic) sampling methods for band-limited signals. In terms of Mean Square Error (MSE), the reduction reaches at least 12dB (fixing the oversampling rate). Additionally, we investigate the quantization process for AIF-TEM and analyze the quantization MSE bound. Empirical results show that classic quantization for AIF-TEM improves performance by at least 14 dB compared to IF-TEM. We introduce a dynamic quantization technique for AIF-TEM, which further improves performance compared to classic quantization. Empirically, this reduction reaches at least 10 dB compared to classic quantization for AIF-TEM.