The data was collected from410 participants and The control variable are age, gender, ethnicity, alcohol use and education. While I have only two dependent variables
If you are using SPSS, you go to regression - linear - choose your dependent variable and than in blocks the independent variables. In the first block enter the control variables (that should be dummy coded) and than depending on your analysis in another bloc or two blocs your two other dependent variables.
There are several methods I would suggest enter, it gives you more information on the data relations and leaves the decision to you. But you could test others as well.
You check after the model summary and report R2 and R2 change
ANOVA table for the significance of each model and the parameter estimates for significant relationships among different variables.
as all regression analysis you have to see the fit criteria first.
This video from how2stats YouTube channel is very clear and to the point, there is also a book available for free, that might be of interest: https://www.youtube.com/watch?v=pkM4nXvP5Bo