Graduate school in engineering admissions question?

I am a senior year Biomedical engineering student at a good but big state university, class of 2021. I came across an unprecedented situation, need your advice.

I’ll be done with my credit requirements (126 credits) by end of this fall semester. I need to complete the design/thesis work to graduate. Unfortunately, our design work only offered in the fall, and to do that, students need to do some prerequisite courses.

Because of scheduling issues, I could not take 1/2 prerequisite courses before, so the university is aking me to take the design course 2021 fall, and I can take internship/co-op starting 2021 spring.

But, in that case, my graduation will be deferred by about a year.
I also took the GRE exam, got a decent score Quant: 168 Verbal: 164 I was all set to apply for the MS/Ph.D. program in Biomedical Engineering so that I can join next fall.

I requested my university department if I can take design calss this fall, but they are not willing to bend rules.

My question is: I’d like to graduate in 2021, then go for MS/Ph.D. programs in 2021.
What are my options?
I am told a good private university may be willing to admit provisionally (as I may not able to get a degree next year).

Is it true? If yes, how do I search & approach this case?

Any other pointers appreciated.

Thanks in advance.

I’m not sure this will be of any help to you but a while back I graduated mid-year from my engineering program (for me it was a study abroad year that mucked up my credits) and joined a different university mid-year as a Ph.D. candidate. I already had a thesis adviser before starting, however, and he advocated on my behalf. I was sad to have missed my undergrad graduation… If you will start a graduate program with no set adviser I would recommend having someone in admissions advocate on your behalf.

Thanks for the tips. Some Questions: Did your old university let you graduate for not completing the required courses? How did you find your graduate advisors?

I did finish all my required courses, just a semester late. I met my adviser during my study abroad year. Good luck!