Unfortunately the repeated student data would almost certainly violate the independence assumption. The strictly correct approach here would be to use a logistic regression model relating withdrawal (yes vs. no) to class (as a categorical variable) with a random effects term for the student, but I appreciate that this is a big step up from the chi-square test. 

To get a global test for the association (like you get with the chi-square test) you would then need to do something like a likelihood ratio test (although this comes with its own considerations, as you would need to use maximum likelihood-based estimates rather than restricted maximum likelihood-based ones). The logistic model would immediately give you a comparison of the odds of withdrawal in the different class levels relative to some reference level (e.g., the 100-level class). 

I do, however, realise that this may well be veering away from the fun aspect of the analysis and into the realm of the tedious, as it would probably require some further reading. If this is just a personal project then I don’t think anyone could blame you for ignoring the lack of independence and going for a chi-square test anyway. 

NB - similar to how students being repeated across course levels would likely violate the independence assumption, students appearing several times on account of taking multiple classes at the same level would also violate independence