I have got a dataset with a dichotomous groupid variable and many other independent categorical variables. My aim is to find out if subjects in first group behave differently than subjects in second group. So far I have done the cross tabulations between groupid and each categorical variable and ran Fisher's Exact Test (chi square test was not valid as some columns have less than 5 value). II want to know as follows.

What are are other statistical test that I can run to establish if groups behave differently.

Could I run Logistic regression with groupid as response variable and all other variables as predictor variables? If the model performs reasonably well will it establish that both groups behave differently? Is this a reasonably way to extablish that two gropus behave differently?