If I start the free trial on Coursera and then end it before I pay, can I just keep watching the class? I do not need the certificate.

The majority of courses on Coursera have an Audit option.

This is not very evident, because to access it you need to click on the free trial button, and the option for Audit is at the bottom of the page which will open. You could log out and visit the course page, to see if the option is there once you click on the trial.

There are limitations to the Audit though. Some tests and labs might not be available, as well as forum access. This can limit the actua learning considerably, unless you look for other ways to test your knowledge, and are lucky enough to find options which mirror well enough the course. It really depends on the course how much of the content is paywalled, in some cases it is very little. But you should at least have access to all videos.

To save cash, I do audit courses and try and learn as much as possible, then if the certificate is useful, I will pay one month and take the exams in that time.

The audit option is a good thing, however Coursera used to offer more for it, which is one reason I stopped volunteering with their translation project.

Lastly, consider that Coursera does offer a means tested free or discounted access to some. However, I am not entirely clear what the criteria are. I assume that someone with no income might qualify.