An agreement, written or verbal, is a contract. You've signed it, so you've agreed to it, and you should look carefully at what you've agreed to.
Generally, the advice is to make sure your employment offer is in writing not because a verbal offer isn't a contract but because it's a difficult contract to enforce.
In the US, it's rare relative to other countries to have employment agreements that bind employee and employer to a specified length of employment. This varies a bit by state and you can come to an agreement otherwise, but in most cases in the US you are free to leave a job at any time and your employer is free to fire you at any time, as long as they don't violate other rules in doing so (like discriminating by gender or race).
However it does say employment is subject to university funding availability and other background processing (criminal record, visa etc.)
That's pretty standard - if you aren't eligible to work at the university because you have no visa, your offer is void. If you fail a criminal background check, your offer is void. Even if you had already started working and were there for a week, if they run out of money they can end your employment, they're only legally required to pay you for the time you already worked.
Now, there is an important distinction between what is permitted by law and what is free of consequences. If you accept an offer and then decide you don't want to work there, the institution can't legally compel you to work there, but they can be annoyed by you. They can refuse to consider hiring you in the future, the faculty there can tell their friends that you skipped on them. If you have good reasons to renege and gave ample notice, it's unlikely those things would happen, though, at least not to an extreme degree. If there are visa problems, they're likely to see the bureaucracy as a shared enemy and not blame you for it, as long as you've done your part. If they were expecting you to arrive and teach courses and you told them 2 days before the semester began that you chose to take another job that's a definitive bridge burned.
Similarly, if an institution regularly hires people to tenure track faculty jobs and then tells them "oh sorry, we decided we don't have the money to pay you, bye" the word is going to get out and they will quickly find it harder to recruit anyone (and likely other people working there will look for other jobs).