Typically you'd stop interviewing upon having a signed contract. Until you do, you don't actually have a job - merely a promise.
But your situation is different.
It seems you were hoping to start right away. Instead of being accommodating, they decided to stall the entire thing until September. That's several weeks away. That's them telling you: "what you need or want doesn't matter; the only thing that matters is what we need." Just run.
It's one thing to tell you that you can start in a week. Maybe even a few weeks, if they provide you with a sane justification. Perhaps they need to sort out some admin BS, like securing a budget to pay for an unplanned extra hire, or making sure someone will be around to onboard you.
Telling you that you'll start "in September" while you're available today, by contrast, is completely different. It's disrespectful at best. At worst it can mean any number of things ranging from "we don't care about you" or "we don't value what you'll be bringing until you fit in this very specific box" to "we actually can't pay you yet but hope we will then, hopefully" of which the main variation is "we told you you've the job but it's actually conditioned on sales closing this deal with that client."
Assuming you've already nudged them to get an earlier start date and they refused, continue to job hunt whether you've a signed contract or not. The worst that can happen to you is you getting more job offers. And giving them an unapologetic "Sorry, I unfortunately couldn't wait until September."
Also, don't forget to set yourself a deadline to decide once you've a few hot leads, let them know when you intend to make a firm decision upon doing so - and stick to that date. (You technically have a date already: the day before you've been told that you'll start.)