Is it self-plagiarism to reuse sources? I've taken two classes in a row that require research papers to be written as an assignment, and since the second one has had a similar criteria for which topics are allowed, I decided to continue researching the same topic. However, after checking with the school's plagiarism checker, it flagged me for roughly half of my 2nd paper's sources being reused. Is this something I should be concerned about? The rest of the paper is completely original and not copy-pasted from my first one at all, plus all of my sources are cited correctly.
In general, no (and how did you get access to your school's plagiarism checker)

If the papers are related they should share sources.  From a published peer-reviewed standpoint, I'd bet several papers reuse half their sources.

This sounds like a paper for a grade though.  In which case you actually have a 3rd option.  Talk to the professor or TA.  Bring the first paper and this paper with you and let them make a decision.  Assuming you didn't plagiarize, they'll see the content is different.

Remember, part of taking a university course is to learn how to write papers with novel ideas.  The professor and TAs are there to help you learn how to present your ideas without plagiarizing sources.  Especially coming before the paper is due shows a good-faith effort.