Flicker (pink, 1/\(f\)) noise is ubiquitous in all electronic devices, including in oscillator circuits used in true random number generators (TRNGs) based on jitter. Flicker noise produces strong serial correlations with very slow decay in time. We present a stochastic model for counter-mode TRNGs that takes the effects of such time correlations into account. Key parameters in the model are the spectral strengths of the pink and white noise components, as well as the low-frequency cutoff for the flicker noise spectrum. The random bits are defined as the least significant bit of consecutive integer-valued count numbers. We present the dependence of autocorrelations and min-entropy of generated random bits on model parameters. The autocorrelation between the bits is suppressed by increasing the strength of either pink or white noise, but it remains long-ranged (power-law decay). The power-law exponent depends linearly on the strength of pink noise, while the prefactor depends exponentially on both strengths. We determine the min-entropy per bit from a careful analysis of long-time sequences. It approaches the value of 1 approximately as a stretched exponential function of the flicker noise strength: highly entropic random bit generators can thus be designed even in the presence of strong flicker noise. We also propose an effective and efficient online health test for generators of this type.