I am writing my thesis currently and I want to include papers that I am included on as an author. I will have a couple papers where I am 1st author, but there is one where I am third but did a lot of the work. How do I go about integrating it into my thesis? Can I copy/paste most of the text still or do I need to rewrite it? I don't want to plagiarize but don't want to rewrite it if I don't have to.
This situation does not sound problematic to me. Many people write theses that incorporate papers written under these circumstances.
However, you need to get official permission, rather than relying on what random people on the internet say. Specifically, you need to check three things:
What does your thesis advisor say?
What are your university's regulations regarding reusing text from published papers, particularly with coauthors where it is difficult to identify who contributed to a given passage?
Does the publication or copyright agreement with the journal permit reusing this text in your thesis?
If any one of these has a negative answer, then you cannot copy the text. If they are all positive, then all you need to do is attribute the text properly. For example, you could clearly state at the beginning of a chapter "This chapter is an edited version of the paper BLAH." Your advisor, university, and publisher will have suggestions or requirements for how this attribution should be done.
Check with your institution. Some universities allow collections of papers to be submitted, provided there is original material connecting them that is also submitted. See, for example, the ANU's "thesis by compilation" rules.
If your thesis is a compilation of your published papers, then check with the institution for the rules.
If your thesis is separately written, then most likely the pieces you wrote can be copied, as that's how I've gone about in my thesis. Additionally, you can also refer to the paper itself after you've written a piece of text about it.
Most likely in this situation, copying the text would not always fit the structure of the thesis chapters, and some minor rewriting is needed. Even after thesis review, the structure might modify and sentences will eventually be different from the paper. You refer having done the most work, but check if your paper co-authors are happy with using the text directly (if they've written the text). If the paper is published, cite it, and you could specifically mentioned that it was your work published in that paper.
Also, your supervisor is there to assist you and he most likely know the rules of your department and institution better.