I took a look at your model. It’s hard to tell what your graph is actually showing though. Could you paste your code for the second visualization? If it’s just showing a distribution plot of all Y_obs, I’m not sure how meaningful it is for figuring out what’s going on. I will say, though, your data seem to contain a lot of zeros and an exponential distribution probably won’t work for modeling counts with excess zeros (or any count data, for that matter).
To answer your question… like @aseyboldt said in your original post, I would recommend using a Negative-Binomial distribution. It’s like the Poisson distribution for count data, except allows you to handle overdispersion.