I am wondering if employers might try to verify that I attended a college and if I need to update them with my current last name as it is different than the one on my diploma.
If your university uses your Social Security Number (SSN), they'll be able to track your academic record regardless of what legal name you are now using. From my recollection of the 1990s, every single university application I ever filled out including for admissions and graduation - that application included a request for my SSN. Foreign students on a student visa are eligible to apply for an SSN, depending on which visa they have (*). Fact is, students get married, divorced, separate all the time including with their professors - don't get me started on that one :)
(*) Thanks for your comment, @BurhanKhalid
Most applications have a place for other names you have used. You simply indicate the name you used when you were in college. If you want to be very clear, and you have space on your resume, you can indicate the name you used in the education section:
BA Degree in Basket Weaving, Walmart University, 2004
(Degree received using name Joe Cool, not Joe Schmoe)
Your problem isn't uncommon. Many people who marry change their surname to that of their partner, and in the 21st century this does no longer only apply to women. So staff managers are used to checking references of people who were once known under a different name.
Just state your name in your résumé as "John Doe (born: John Smith)". That way it should be obvious to everyone that John Smith and John Doe are the same person.