What can I do if the terms of my adjunct contract are not what I was led to believe they would be?

First of all, consult a lawyer before doing anything.

If you were to leave in the middle of the course, that will cause havoc, that they won't probably want. Since they are not paying you for the work, you are under no contractual obligation to fulfil anything (but consult the lawyer), and can leave without giving them notice.

Now, you may find unethical to leave your students hanging like this, but the mere threat that you are willing to do it, may get some wheels spinning to give you a decent offer.

Another option to consider is whatever anti-slavery laws apply in your jurisdiction, since you are working for free; and anti fraud, because you are working without a contract. Furthermore, since you don't have a contract, you are not covered by the insurance, and if anything were to happen during your lecture, there is trouble. Think who would have to pay for in case, for example, a projector caught fire and ruined the classroom.

If your oral agreement is considered a contract, you can't do this; but then they would be acknowledging they have to abide by these rules.

The fact that they did not pay you for two pay periods means that they have committed wage theft in most states in the US. You should go to an employment attorney immediately. You almost certainly have an oral or implied contract, and whatever they are saying about you not having one or "rules" requiring one is malarkey. You worked, under terms that they agreed to, and they owe you back pay and future pay if you continue to work. I'm not a lawyer nor am I your lawyer, so go get one. You are under no obligation to continue to work, but I'd guess that if you do, you will help them dig their hole deeper, since they will, at the very least, continue to not pay you until they've heard from your attorney, your state's Department of Labor (or equivalent), both, or you've won your lawsuit.

This institution seems to be taking advantage of you and not treating you with respect. I honestly think that you should quit and try to find opportunities elsewhere. If not in academia, in industry. I assume that you are a Physicist from your profile, and they are increasingly being hired in the software industry, so perhaps a change in career path? If not, this institution is at least in my opinion, not valuing you as an individual, much less an employee, I would advise moving on. I know this may not be practical, but having been engaged in research in academia and advising, I can honestly say that it may not get better.