If the outputs are highly sensitive to the prior it means the data are not informative about this parameter. This is likely an identification problem. There are no simple ways to diagnose these types on problems, because they are model specific. To paraphrase Tolstoy, every mis-specified model is mis-specified in its own way. Here is a blog by Michael Betancourt on the subject, but it’s not exactly brief.
Looking at pair-plots can be interesting in these situations. You might also consider mapping out the gradients of the likelihood function for a range of values for this parameter, setting other parameters to their posterior means. My expectation is that they will be close to zero.