Thanks Junpeng! I tried lots of variations: larger and larger sds for the priors, less and less regularizing hyperpriors for the sds (half normal, half cauchy). Also changed the priors themselves (StudentT or Cauchy instead of Normal). To no avail: the estimation still seems strongly biaised.
If I understood correctly, in the new parametrization, you’re using all the regressors right (unemployment as a random effect, the others as fixed effects)? Do you think the biais can come from there - maybe there are too many regressors?
Also, the data actually contains another type of cluster: the type of election (presidential, parliamentary, etc.). Do you think modeling it can reduce the observed biais?
Or maybe does it come from something else entirely? 
(unbiased in the way that if you computed point estimate by just taking the mean it is at the mode of the posterior, also I should mention that the multinominal model i was working with only have categorical predictors - 2 group of subjects doing 3 very similar task, so I have group and task as fixed effect and subject as random effect). Here the code above is the best model i concluded. Maybe you can try removing the continuous predictor and keep only the categorical ones and see how the model performs?