Just realized I may have accidentally plagiarized some of my bachelor's thesis I had been reading a news article recently about some politicians who had been accused of plagiarizing their master's or PhD theses, and I noticed something that really frightened me.

With some of them, what they did was copy, near-verbatim, passages from other articles or books which had citations in them, and they cited those same citations in the footnotes, without citing the secondary source. FWIW, those officials did so in humanities, law, and/or international relations. In contrast, my thesis was in a STEM field.

I was unaware that this is considered plagiarism, and realized that I had done something similar in my bachelor's honors thesis. I had looked at the thesis of a former student in the lab to get an idea of how a thesis was written. His work was closely related to mine, and as I was short on time, I ended up writing an introductory and methods section that were most likely, in some sections, extremely similar in wording and style. For the introduction, I tried to paraphrase what he wrote and cite the same sources he did (I did check to see if the sources were accurate) on background information relevant to both of our projects, but I'm sure that a side-by-side look would spot the similarity, as the sections were mostly definitions and descriptions of things we work with.

For the methods, as I had used some of the exact same protocols and procedures that he did, I just wrote down what I did but the wording most likely ended up being extremely similar to his, enough to notice, with no citation given for that part because the protocol was just given to me orally by the grad student when I was starting out. I thought this wasn't an issue because the methods section is difficult to phrase in different words if the experimental protocol was the same. I passed the defense and the thesis was approved - my PI didn't notice any similarities. 

The parts of the thesis where I branched off in a different direction than that student I wrote without the aid of the previous thesis, and I cited sources as appropriate and reported my own work.

My former PI stores hard copies of his students' theses on a shelf in his lab, and there is another hard copy in the university library, but there are no electronic copies available. I don't have the thesis of that former student with me. What I'm terrified of is that a future undergrad in the same lab might use my and that person's thesis as guides to writing their thesis, notice the similarities, report me to my former PI and/or the honor council, and get my degree revoked for plagiarism. It's been a year since graduation and I'm about to head into grad school.

Do you think I am right to worry? If so, what should I do?
This is an appropriate thing to be concerned about and want to rectify; you don't need to fear dire consequences, especially if you take steps to address it. There are different kinds of plagiarism, and also different levels of awareness of what one is doing. Turnitin has an interesting diagram with catchy titles to distinguish them; yours sounds like a mix of &quot;hybrid,&quot; &quot;recycle,&quot; &quot;aggregator,&quot; and &quot;retweet.&quot; Probably the hardest overall concept to understand--and one that may not be fully comprehended by many undergraduates--is that when a key source shows you the way to other sources, you need to be careful that your paper isn't entirely dependent on that key source for its content and/or structure. You should also make sure to acknowledge if a quotation in your paper from a third source is there by way of that key source.
I tend to agree with Anonymous Mathematician's answer to a similar-ish question about plagiarism in a doctoral thesis. In particular, identifying the problem and issuing some sort of errata or revision, prior to anyone else coming across it, is likely your best path. Making a mistake as an undergraduate, and recognizing it and working to rectify it, is unlikely to pose problems for you if it later comes to light. (This is largely the tenor of the answers to a question by an undergrad about publishing in a journal, realizing later that the work contained cited but unquoted direct quotations.)
That said, it would be wise to seek counsel from an ombudsperson at your undergraduate school and your graduate school, as well as your undergrad advisor and your new advisor. You don't want anyone to be surprised by anything and jump to the wrong conclusions. Exactly how to handle the process of corrections might be delicate, but it is certainly the right path to take. The worst case scenario would be if you confessed this to your undergraduate institution to address it, and they were required to report this to your graduate school, which only read the words &quot;academic integrity issue&quot; and rescinded your admission. With proper communication and feeling things out with ombudspeople at both institutions (confidential advisors who can help you navigate the system), you should be able to avoid such a misunderstanding.
Minimally, you should probably clearly and concisely acknowledge your reliance on the other thesis, perhaps as an erratum sheet you could send to your undergrad advisor to stick into a copy of your thesis. It might say something like:

Much of the literature review and methods sections rely on the work of Prior Student's thesis, and I would like to properly acknowledge that debt.
I found sources A, B, C... (discussed on literature review pages X-X, Y-Y, and Z-Z, respectively) through Prior Student's work.
Prior student did similar experiments on __, and I relied on their phrasing for much of the content on pages X-X.

In another approach, you could produce a revised version that somehow does not rely on the previous student's work, and mention in a note that you produced the revision because you realized that you had inappropriately relied too closely on the other student's work. (For instance, find several more papers, read them and reread the sources you had before, and write a new literature review.) However, if you expend this effort without any guidance, you might not distinguish your new work enough, and/or you might be moving toward a different resolution than your advisor and undergraduate institution would find acceptable.
Good luck!