Doing separate ANOVA for each predictor ("independent variable") is not a good idea. You run a risk of omitted-variable bias if the predictors are correlated with each other and with outcome. All of your estimates for associations of predictors with outcome then might be incorrect.
A standard linear model would evaluate all predictors together at once, taking each other's values into account. The R lm() function can work well for this. It can also work with both of your outcome ("dependent") variables together in a true multivariate (multiple-outcome) model. See this article, for example. This UCLA website has links to how to perform such a "multivariate multiple regression" with 3 other software packages.