I think generally in this situation doing independent sample t-tests is reasonable.

The point estimate is the same (ignoring missing data). One *typically* gains power by doing paired sample t-tests (when the measures have a positive correlation, the standard error of the difference is smaller). So given that assumption (people who test high early are also more likely to test high later), that standard error of the difference for independent samples will be larger than in the paired sample scenario (if you want to be really conservative choose the sample size N to be the smallest of the three samples).

Sample attrition is likely a bigger deal than worrying about the correlation between tests in this scenario. E.g. if people are more likely to take the later test if they know they will do well, or people who did poorly were more likely to drop the course. This is something that I believe is unsolvable with the information provided, so is a fundamental limitation of the analysis.

(Maybe you could come up with reasonable potential bounds on the bias though given the sample attrition? I am not sure.)