I have been offered an apartment. How do I accept?

Apologies for what is probably a neurotic question but how do I officially accept an apartment? I received an email "Zusage" from the management saying that they wanted to offer me the apartment and that I have to send them a binding commitment. Specifically, the email said:

,,Damit wir den Mietvertrag per 01.08.2023 erstellen können, bräuchten wir noch eine verbindliche Bestätigung.

Besten Dank für Ihr zeitnahes Feedback.''

I replied thanking them for the good news and that we are happy to commit to accepting the apartment. Unfortunately, I didn't ask about the timeline for receiving the rental contract.

I sent that email a few hours ago. Should I send another asking for an ETA on the contract? Should I have responded differently?

I'm just super nervous about messing up this process. Thanks in advance for answering my anxiety driven questions.

I think you did just fine. I think I'd wait until tomorrow, and if you don't hear from them, ask when you can expect to receive the contract.

That's what I'll do. Thanks for the reassurance.