Underdispersed binomial/overdispersed beta-binomial

Hi! I’ve got census data and I am trying to model Education (college graduate or non-college graduate) from four predictors (US state, sex, age, race/ethnicity). Since I’m trying to model a binary thing, I first tried a binomial model and various combinations of the predictors and interactions. In all cases, my data had higher variance than the binomial model implies so, e.g., I get shinystan ppCheck pictures that look like:
And these fits have terrible LOO, almost a third of observations have Pareto_k over 0.7.

So then I tried beta-binomial with various parameterizations for the “dispersion” parameter. These are happier in LOO terms but they are now the reverse with dispersion and the means are off:

I’m a little flummoxed! Are there some obvious places to look/things to try here?

Edit/Update: I’ve tried parameterizing the dispersion so I can force the beta-binomial to be more like the binomial. I thought there might be a “sweet-spot” where the mean & dispersion of the posterior prediction would be a better match but I would not lose the better quality of fit as indicated by LOO. That doesn’t seem to exist. Well before I go from over dispersed posterior predictions to under-dispersed (like the Binomial), I start getting very bad LOO results.