We numerically and experimentally analyze the output characteristics and pulse dynamics of carbon nanotube mode-locked fiber lasers near zero cavity dispersion (from 0.02 to ~-0.02 ps(2)). We focus on such near zero dispersion cavities to reveal the dispersion related transition between different mode-locking regimes (such as soliton-like, stretched-pulse and self-similar regimes). Using our proposed model, we develop a nanotube-mode-locked fiber laser setup generating ~97 fs pulse which operates in the stretched-pulse regime. The corresponding experimental results and pulse dynamics are in good agreement with the numerical results. Also, the experimental results from soliton-like and self-similar regimes exhibit the same trends with simulations. Our study will aid design of different mode-locking regimes based on other new saturable absorber materials to achieve ultra-short pulse duration.