Normally you'd inform your manager verbally and then hand in the notice in written format to them + HR, usually this is a more or less brief "terminating contract by date X, please confirm" and that's it. HR would then get back to you confirming the last day of employment and can calculate outstanding leave, etc.
I was let go yesterday from a job that I was supposed to start in January. I had done all the "right" things, waited to sign the new contract before handing in my notice and everything. It didn't do me any good, there's no security, although in most cases, yes, people will tell you to not resign before having something new lined up. It really depends on your area and also how quickly you can find something new, your financial situation, etc. But - no guarantees. I haven't had the case that someone got a bad reference out of spite/not wanting them to leave.
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I'm so sorry to read this, it sounds indeed like you were sold a job that essentially doesn't exist, or that was simply taken over by someone else and it is leaving you with the bits they can't be bothered doing. As harsh as it sounds, it seems unlikely to me to change - if you are asking for work and aren't receiving any, plus the social dynamics in the team also leave you out, I'd take that as the sign to start looking for something else. You have tried, you have indicated preferences on how to do something, it seems like you got unlucky by hitting a team that is unwilling to accommodate you and accept some freshness and that might never have even needed more than one person ❤