Should I stop applying faculty positions after I take on a postdoc job

Don't stop exploring other positions until you have a firm commitment from the professor. The situation is mutual. You both have to agree before anything is finalized. Until then you are still free to consider all options.

By firm commitment, I mean something that you can put your signature to.

Don't stop exploring other positions, as long as you give the current prospective position the same amount of attention as if you weren't applying for other positions.

Remember that the academic hiring cycle is usually several months long. Applications you submit now will land you interviews in the spring, with jobs starting in the fall - nearly a year from the application time! If the professor is only willing to give you a one year commitment, you may need to start applying for faculty jobs this cycle to have a job lined up when their commitment to you ends.

In that time period it's also to your benefit to work hard turning the crank and getting out publications that you can show off during your on-site interviews. Thus, your goals (to get publications to land a job) and your prospective boss's goals (get publications out of the postdoc for the next grant submission/tenure review/...) are well aligned. Also keep in mind that postdoc is a training period, not just a purgatory you must suffer through to get the faculty position. Presumably you're taking this new position because you believe this professor has the ability to help you get high-level publications and can teach you something about how to be a successful faculty member. Amongst all the applications, interviews, research, etc., don't forget to make the most of your current position and learn what you can from this prospective supervisor.