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. With multivariate modeling you get the same coefficients and standards errors as if you did regressions for each outcome separately, but you get more reliable comparisons between the two outcomes.