Statistical test for non-normal, dissimilarly distributed data?

I'm trying to determine if there is a significant difference in specific-leaf area values based on two categorical soil type values. Here are what two of my distributions look like:
Screen Shot 2021-09-16 at 3.27.50 PM.png Screen Shot 2021-09-16 at 3.28.04 PM.png
I'm thinking of using the Mann-Whitney test but I have also read it is not the best for differently shaped distributions. I also have various sample sizes, some are below 30 per group, others are above 400 per group. Also, the majority of my data have unequal variances. These categories are unpaired. Is there a better test, or tests, that I can use for these types of data? Thank you.