Could these DUIs affect me petitioning for a K-1 Visa?

Although not entire clear from your question, I presume that you are a U.S. citizen and that you are interested in sponsoring your fiance to come to the U.S. to become married to you within the required time, and then to sponsor her for the adjustment of status application process by which she would seek to become a Lawful Permanent Resident (to get a "Green Card"). If that is correct, then be aware that a U.S. citizen's own criminal background generally is irrelevant in the application process, and so your DUIs and drug possession charge should not stand in the way of sponsoring your fiance for immigration benefits. Of course, many other details have an impact upon immigration eligibility. Beyond that, the application process can be far more complex than it might appear just from reviewing the immigration forms, reading the immigration regulations and reading the USCIS website. It would be wise for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.