Irregularities in the contract job process and approach

Got an offer from one of the top professional IT service companies for a FMCG giant asy client on 3rd party payroll. The role is the exact fit according to my career trajectory. Everything happened within a span of 5 days. Tried bypassing by applying directly but the posting was removed the day I appeared in the first round of the interview. Connecting, cold mail, nothing worked for direct application so I was considering getting a chance through this route. But what's surprising, I was informed within 20 mins of the final evaluation that I was selected - 6 months contract. I need to immediately report at the office in a week - 1.5 months and then I can ask for WFH. Post that contract period, I will be working in the company payroll directly. I have to source a laptop for work, 3P payroll will reimburse the amount, health insurance will be provided by the 3P payroll - said the PoC. When I press about the contract letter and terms, she keeps saying that it's a full time opportunity like I'm slow. I had to clarify the definition to her. All of this completely verbal exchange. They're in a hurry to just get over the formalities. I have documented things clearly to avoid missing anything and they should release the contract letter tomorrow.

Is it even advisable to relocate and work in such a situation?