Overdispersion and over-abundance of zeros are two different situations that are modeled in different ways.
Overdispersion refers to a variance that increases faster than the mean. For example, a Poisson assumes variance equals mean, and thus increases at the same rate. If variance in fact increases twice as fast as mean does, then the Poisson is not appropriate because of overdispersion. Overdispersion can be picked up in NB and Quasi-Poisson models, for example.
Over-abundance refers to a greater number of zeros than assumed by traditional distributions, like Poisson or Negative Binomial. You can simply fit said distributions to a histogram of your data to see whether you have over-abundance of zeros. You can reflect over-abundance in zero-inflated models, like ZIP and ZINB. You can also consider Hurdle models, but note that the assumptions as to the sources of your zeros differ between Zero-Inflated and Hurdle models.
My suggestion would be to first determine which of these situations is affecting your dataset and proceed accordingly.