This is a Troublesome Pair that I find very commonly in academic writing, across a whole range of writers. It would be expected in people whose native language doesn’t use vowels (I had a very interesting conversation about Arabic-speakers having trouble with vowels in English) but I also find it in native English-speakers (it’s actually fairly uncommon to find both groups making the same errors)
To adopt something means to take it on as it is. For example, Laura might adopt my process of editing a text, then doing a spell check, then using software to check consistency. If the UN adopts a resolution, it means it takes it into its procedures as it is, with no changes. If you adopt a dog from a shelter, you take it as it is.
To adapt something means to change it according to specific circumstances. For example, Laura might adapt my process by choosing to edit the text, then run the consistency software, then run a spell check. The UN might adapt a resolution to take into account a new world order or the creation of a new country. You might try to adapt your adopted dog’s behaviour if you don’t want it to sit on the sofa.
Adoption therefore means the act of taking on something as it is, while adaptation involves you changing, or you changing something else, to fit the circumstances.
In academic writing, a researcher might well adopt a questionnaire method, but they might adapt someone else’s questionnaire if it needs more questions on widgets and fewer on bath mats, for example.