No, you absolutely will not be required to pay anything yourself.
My mother was self funding in her care home for several years, and her funds got down to the £23k threshold last year. During the financial assessment Social Services asked if I could provide a top-up, I said no and that was the end of the matter. The local authority now pay her fees (with her pension is taken towards them), and my mother is still in her original care home.
When the time comes, you contact Social Services and fill out the forms (I did this via an online assessment tool they sent me). You then have no further input. The negotiation of the level of fees happens between the care home and the local authority.
When her funds were getting near the threshold, I went into the care home and spoke to the Financial Director and he explained to me how it worked, and I kept him updated with any contact I had with Social Services.