Getting a F1 Visa After an Overstay

I am an F-1 student whose SEVIS record was terminated a few months ago. I let myself go out of status because I got married and thought my wife would get me a green card. Well, my marriage didn't work, my wife won't sponsor me for a green card, and now I want to go back to school. My school readmitted me, but I can't do F-1 reinstatement because I can't afford to wait. So I was told that my only other option is that I need to go back to my home country to get a new F-1.

I was told by my international student office to bring evidence of my marriage and that it was genuine, because that's why I let my status terminate and without evidence of a bona fide marriage, I wouldn't be able to justify why my status was out.

I was told that I might not be able to get my F-1 because I terminated my original status or maybe that getting married to a US citizen shows immigration intent. I don't have much proof that we were married to show anybody.

What kind of trouble am I in? Is it likely my F-1 application will be denied? Any advice is requested.

If you were only out of status 90 days and left on your own, you probably don't have any time bar on readmission.

However, you have a BIGGER issue. You have immigrant intent being married to a US citizen residing here. You will likely be declined on that basis alone.
It's unlikely you're coming in to the US period without your spouse agreeing to be responsible that you won't end up on the public dole.