I am running a full factorial anova on my data but when I apply Levene's test and Shapiro test not all of my subsets give positive results. I have already transformed the data with log2 but that doesn't change much about the homogeneity or normality. Someone told me that I might be able to use a type 3 Anova instead but I can't find any specific information about whether you can or cannot use type 3 for not-normally distributed data. Can I use a type 3 anova do I need to do non-parametrical tests for the not normally distributed subsets?
Using a type-3 ANOVA will not eliminate the problems associated with heteroscedasticity or non-normality. While it may potentially give you greater power to detect the effect you are interested in, this is conditional on meeting the ANOVA assumptions in the first place. In general, the choice of the 'correct' type depends on the hypothesized effect (main, interaction) and your cell sizes and may thus be effectively considered fixed.
It would be helpful to have the following information: a histogram of your distributions, the group variances, the group sizes, and possibly the outcomes of your tests. Even if Levene's test and Shapiro fail, it might not be strictly necessary to switch to a Kruskal-Wallis test.
As for transformations, R's boxcox function allows you to explore a range of values for your transformation parameter.