Verbal offer but no contract, still feel in purgatory. Starting my career, unsure what to make of all of this

I have been back and forth at a big large media conglomerate company. First my old supervisor recommended me for a position I applied too. Here is the timeline.

6/15: screening talk with the recruiter at company. Basic questions.

6/17: phone chat with who would be my future boss. Talk goes well. Interview got delayed bc he was trying report on important news coverage (yes, job is in news). He tells me to take a writing test.

6/22: he sends me the writing test, i hand it back to him within three hours. He tells me that if “the news allows him” he will get back to me before Friday, then if not send a note. Friday comes, I send a note.

6/27: hear from the recruiter, i passed the writing test and i have two more sets of interviews with the rest of the team. Two back to back. They went well.

7/6: i follow up with recruiter because it has been silence. This was after holiday weekend so i knew it was hectic, she told me Monday she will have an answer.

7/11: recruiter CALLS me & tells me i did not get the first position but they like me so much to watch out for another position and posting. The “future boss man” was on vacation this week so she tells me communicating with the team has been chaotic. I don’t take the thing as a L and think its good the recruiter likes me.

7/15: i get an email from one of the managing editors which interviewed me on 6/30 saying they wanna zoom to followup. She offers me a similar position with the team just different hours. Same pay. She tells me that she didn’t know the recruiter reached out to me to first tell me I didn’t get the job. Sorta red flag?

7/18: i accept the job Monday and email her telling her and she tells me its great, my future boss is now back from vacation and will help me with next steps.

Literally not a peep since and that was Monday. I was told they are hoping i can start beginning of August. I was going to send a followup email Monday just asking for a timeline. I presume they were busy with the olympics and such/playing catch up but this push and pull is getting ridiculous. I need a contract to sign already. I feel like nothings really changed. What is the right thing to do here?

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