I moved to a new city and I've spent last weeks looking for a job intensively.
After a long application process with several interviews over several weeks (it's a temporary position, I was surprised the process took so long and was so complex), I got an offer last Tuesday around noon. They told me they wanted me to start on Feb. 13th (this Thursday). As I was in several other recruitment processes I asked for a few days to make up my mind. They told me they needed my decision by Friday early morning given I would be starting shortly. I called them earlier than that, on Thursday morning telling them I wanted the job and starting on Feb 13th was OK. Additionally I wrote them an email. They told me to wait for the contract to be sent immediately.
At this point I still didn't have any written offer, nothing in writing, just what we verbally discussed.
I waited on Friday and yesterday, didn't want to be impatient. Nobody contacted me. So yesterday afternoon I contacted them by mail asking for an update since I still haven't heard back concerning the contract. Got no reply. Today in the very morning I needed to call them twice. They didn't call back after the first call. The second time the reaction was: "You are starting on Febr. 13th. Oh, it's this week? So sorry, let me check it". They were so surprised I got the impression that if I hadn't contacted them (several times...) they would have never contacted me again.
It's just now that I got the information I can start next month as they haven't got some signature. The HR told me they could try to keep the date Feb. 13th but given that in my country you need to do some formalities before starting and normally need at least 2 days to arrange them, that's totally unrealistic.
I now got invited to the last rounds by two other companies offering the same or better conditions and I'm not sure whether the process at the company described above was a red flag or not. Would it be unfair if I now ask them for a week or so to sign the contract since I want to consider other opportunities?
It's a white-collar job at a big company.