Distribution that acts like Poisson/NegBin for small means and like a Normal distribution for large means? I want to generate a full density probabilistic forecasting model, where I don't know a priori whether the time series I want to model are intermittent or dense. In both cases, the time series is a count series (i.e. I would rather not have non integer outputs - but I will settle for them if there is no other options).  
I know that for intermittent time series, Poisson or Negative binomial are the recommended distributions, and it seems that for dense time series, everybody is just assuming a normal distribution and moving along. 
In my case I want a distribution that covers both and is cheap to calculate and parametric - using some non-parametric approach is too computationally expensive (and too painful to code). 
My first thought was to simply go with Poisson or NegBin, since for large means, they tended to look like Normal distributions anyway. But then I realized that both of those come with a fixed variance for any given mean, and I want a distribution that can be narrow or wide based on the particular time series in question. 
Is there any distribution that satisfies all of my requirements? 
 A: I am not aware of specific work in this direction. You will likely need to "roll your own" approach.
I would indeed use a NegBin distribution, with a time-varying mean for each series (whether you want to model this separately for each series, or connect the series in modeling via a hierarchical, panel or other approach is a separate question).
In terms of the variance, you could use a separate variance for each time series, which is fixed for that series. That would indeed give intermittent behavior for low volume series, and "normal-like" behavior for high volume ones (still discrete, though). Each series would have a modeled spread appropriate to its history. I did something like this (Kolassa, 2016, IJF).
More exotically, you would also model a time-varying variance for each series. For instance, variance is often higher in high season than in low (although this could already be captured by modeling overdispersion instead of variance itself, i.e., modeling the variance as a multiple of the mean, and treating this multiple as constant over time). You could do some thing like first modeling the mean using a NegBin time series, then use GARCH or similar for residuals. Or potentially use a state space framework that might allow you to model states for both mean and variance. Unfortunately, I do not know of any published work in this direction.
