For example, if I lived in my home state during the summer, but on campus in another state (grad student, non-dependent, older adult) for 4 months in spring and 4 months in fall, how do I enter that into the form? It only allows a single start and end date.
Where do you consider home? You can have only one domicile. I guess it is telling that you said "home state".
Your domicile is where you pay a lease or mortgage for a home, have a landline (if you still have one), have your drivers license, are registered to vote, tell your friends where you live, where you plan to return to when you return from a job out of state, and so on.
Working out of state, even if you rent an apartment, is not your domicile if when the job is done, you plan to return to your "home state".
Based on what you have said, I would say that you are domiciled in your home state, thus this is the state you file a resident return for. You live in that state all year, because temporary absences don't count. However, since you are working in another state, you would file a non-resident return for that state where the college is.
Please note that some states require you to file a resident return even if you are not a resident (your domicile is elsewhere) if you are in the state for more than a certain number of days (for example, Virginia says 183 days).
However, since you have not mentioned either your home state or your "college state", I can't say more.