Is an official signed contract required to finalize a work permit and/or residence permit?

Hello,
I've been working for a large English training company in Suzhou for 5 weeks. Work permit has been obtained. Last week, I went with the recruiter to some crowded govt office (police station?) and dropped off my passport for what I think is the final step in transferring my residence permit from my previous employer.

However, I've yet to sign a contract. They had me wait indefinitely while headquarters were making company-wide edits. Supposedly, it's coming shortly. So far, I have only digitally signed a screenshot of the offer letter. The letter uses only English, is titled "offer of employment", lists the compensation basics, but does not mentions working time or number of hours.

Does this necessarily mean that the company has submitted a false/invalid contract to the government? Where could I go to confirm this? I read an employer must pay double salary for every month worked without a valid contract. Where would I go to apply for this compensation?

Thanks for any help.