AST expiring in two weeks, what happens if cannot agree on new rent price

We're about two weeks away from the expiry date of my lease (a standard AST agreement, 24 month with a 12 month break clause). Having not heard from my landlord, I reached out regarding renewal and she said happy to renew but is suggesting a new increased rent price that I don't feel comfortable with.

If we can't come to an agreement on renewal terms, do I have to leave in two weeks, i.e. by the expiry date of the lease? I've read some rules like Section 21 which requires the landlord to serve a notice to vacate at least two months in advance -- is that right? If so, and assuming they serve it tomorrow, would I be able to stay in the flat until Oct 6?

Would love to get some guidance as I am fairly new to the UK.