For ordinal categorical response data, the best thing is usually to use a cumulative multinomial logit model. Those are somewhat non-trivial to implement, though I think SAS can do it and there is (I think) some R package that can do it. I would have to check on which one though. The idea is that you're modeling the probability that the response is less than or equal to each category.
For the dependent variables, you can keep those as categorical, or you can set them to be ordinal values and use a linear trend on your selected values. The latter is usually effective in practice even if the interpretation of the coefficients is challenging. For a logistic model, the Mantel-Hansel test will let you test the linear trend. For a cumulative multinomial logistic model, there might be an analog that works as well.