@Bma009 - Many graduate ME programs do admit students who did not graduate with an ME degree. They will require what are called “leveling” classes in addition to the regular graduate classes. But you will need to be reasonably close, so that you do not have to take too many classes. You should certainly have already completed all your calculus, linear algebra, statistics, and physics courses in college, so that leveling courses could be limited for example to thermodynamics, statics, and mechanics. Also, look into enrolling as a non-degree student in your local public university during the summer or a gap year to pick up the required courses.
Look at it from the point of view of the graduate school when comparing two applicants. Applicant one has a solid but not spectacular GPA in ME. Applicant two has, like you, a great GPA, but has not taken many of the classes that Applicant two has taken. Both have great GRE scores and recommendations. Who is most likely to succeed as a degree candidate?
Here’s what Texas A&M has to say about leveling: https://engineering.tamu.edu/mechanical/admissions-and-aid/graduate-admissions/leveling-courses.html (I use TAMU as an example because my son graduated from there with an aerospace engineering degree, and I had looked into their leveling requirements before for another potential grad student.)
BTW, this topic has bee discussed before on this board.