For your first question, ORs are ok for cross sectional studies (in fact, cross sectional studies mainly use only ORs).
However, cross sectional studies themselves are not very strong evidence of whatever it is you're trying to show. This is because it is only a snap shot of what is presently going on and doesn't allow for any causal inferences.
For your second question, one possible thing you can do is collapse. If you cannot collapse, then I would suggest showing your data to someone who knows what they're doing.