Week before Master's thesis deadline, found publication that explores same concept with slightly different results

You will need to speak to your advisor about this matter, but if the paper is relevant to your research (and it certainly sounds like it is) then you are correct in your view that it should be incorporated into your thesis. If one week is insufficient to do this, I recommend you seek an extension for submission from your supervisory panel. It should not be too difficult to secure an extension in this circumstance, and it is likely that your panel will want you to do the extra work to incorporate this work into your thesis. The golden rule of all research work is to make it as useful as possible for the reader. Adding reference and discussion to this other paper sounds like it will certainly improve your work and give relevant information to the reader about the state of the field.

Finally, in regard to the "horror" you are experiencing, it might aid you to learn that several highly aclaimed researchers have had similar ---but far more horrifying--- experiences of this kind. The most famous case I am aware of is Gottlob Frege, an eminent German philosopher and logician, who wrote his Magnus Opus on logic and arithmetic, only to be informed by Bertrand Russell that one of the core axioms on which the entire theory was based was inconsistent (and therefore the entire theory was invalidated). It is instructive to read the history of this matter (see e.g., here), and it is a fascinating story in the history of mathematics and academic publishing. Russell later remarked on the immense integrity of Frege in dealing with the disappointment of finding out that his life's work was wrong:

As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known. (Quoted in van Heijenoort (1967), 127) (Quote taken from the Stanford Encyclopedia of Philosophy, History of Russell's Paradox)