Could depression/anxiety cause a lack of common sense?

It could. It could, severe depression can lower your cognitive functions, .
Well, your focus/ Attention ; problem solving skills may not be as good as normal. You could also have a drop in your confidence level regarding making decisions and in reacting to matters. Sleep deprivation ; distractability can also take their toll. So, in an indirect manner both depression and anxiety might impact your common sense. Take care.