A sigma of 10 can still be a lot on the exponential scale, especially for slopes. I think the best solution is to do prior predictive checks, to choose the priors most appropriate to your use case.
Here is a NB I wrote on how to do that for multinomial-softmax regression, but you should be able to understand the concept and adapt it.
Also, did you standardize your predictors? This often helps sampling – and even the results interpretation.