Selected for Job in July, still not written offer letter

Hello-

I was selected for a job in July for a city health department. Since that email, I have been reaching out to the HR person to get more info on a start date, pay and etc. Last week was the first time I received a call about pay and I asked to negotiate if possible. They stated I'd hear back within a day of that call but it has been a week now. I'm not sure if I should just give up on the job since a similar position has opened up where I live.

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