That's bizarre - as I posted earlier on this thread, the earliest date offered to me when I received my Lime Tree text today was 21st November.
I'\ve just gone in again and it's still the earliest date available.
I had my national NHS invite for the booster a few weeks back. As you say you have to go to either Brighton or Chichester for them.
This morning I had my invite from my GP surgery (Lime Tree) to book with them in Worthing. The first available appointment is the 21st November.