Usually in designs like yours where you would use multiple categorical predictor variables with a categorical DV (dependent variable) you would indeed not go for ANOVA (since this requires a higher measurement level for your dependent variable), but 'loglinear analysis' instead is commonly used for this and will be appropriate*appropriate here.
Hope this helps and good luck with your reseearchresearch!
*(However, I am not 100% sure you can run this (e.g. in SPSS) with two DV's, and for hypothesis testing simply running two separate tests might not be perfect as you double your chances for a false positive, depending on your exact hypotheses. So that is something to keep in mind)