My first choice would be WTX2WY's recommendation to try to fit the classes in sooner and graduate in December. That way you could start your hike whenever you want in 2021.

If that's not a good option, and you have to take them that spring, why start the trail in March? March is the most crowded month, and the weather will be unpleasant for much the first several weeks. On the other hand, starting the third or fourth week of April or even the first week of May would mean your hike and your class would barely overlap at all, since most college courses are finished by the first week of May. You might have to do a little bit of work from your motel in Hiawassee and Franklin, but your classes would definitely be over by the time you hit the Smokies (where there are not so many convenient spots to hop off trail and find an internet connection).

I started NOBO April 13th and if I was going to do it again, i would probably start even later. I only dealt with a few sub-freezing nights at the beginning of the trail, and I finished the first weekend of October when Maine was absolutely resplendent in fall colors. Late April is a good time to start anyway, but especially if it means not having to worry about your online classes for 4-6 weeks like if you started in March.