You are correct in the essentials.

This is a more complicated scenario because you also have the potential that some of the independent variables are correlated with each other. This is called multicollinearity. Create a correlation matrix between ALL of the variables and see whether any of the DVs are correlated with each other. You could also run a multiple regression and check the VIFs (Variance Inflation Factors). VIFs > 5, especially > 10 show a high degree of multicollinearity.