"But then what do I say?"
Here's the fear that makes all of this hard to act on. Suppose you stop drafting and listen to the very end with nothing prepared. They finish and look at you, and your mind is blank. You're afraid you'll have nothing to say, and that you'll seem slow, or uninterested, or stupid.
The fear is reasonable, because drafting has been your method for a long time, and giving up a method feels like giving up the ability. But it rests on a false picture of where replies come from. You've assumed they have to be manufactured in advance from your own stock of advice, stories and opinions. There's a much nearer supply, which is what the person has just said.
If you've really listened to the end, you're holding their last sentence. It nearly always contains everything you need for your first response. There are three easy forms that response can take, and none of them needs any preparation.
The follow-up. Ask about something they just said. "What else has been a bit much?" "What did Beth say when you brought it up?" "How did you feel when he said that?" A follow-up can't be drafted in advance, because it depends on hearing the end, and so it's the clearest sign you can give that you did hear it. It's also easy. You don't need any wisdom or experience to ask one, only some curiosity about the last thing they said.
The reflection. Say back, briefly and in your own words, what you took from it. "So it's not really about the hot water. It's that you've got no space that's yours at the moment." You might be wrong, which is fine. If you are, they'll correct you, and the correction moves things forward. "No, it's more that I feel like a guest in my own flat." If you're right, they feel the particular relief of having been understood, and they usually go further.
The plain acknowledgement. Sometimes the honest response is simply to register what they've told you. "That sounds exhausting." "Oh, that's wonderful." "I'm so sorry." Said with attention, after a beat, it's often all that's wanted.
None of these is a technique that belongs to therapy, though therapists use them. They're what interested people naturally do. You do them already, in conversations where you happen to be fascinated. When someone tells you a really gripping piece of gossip you don't draft a reply. You say, "No! And then what?" All the method asks is that you do deliberately, in ordinary conversations, what you do without thinking in gripping ones.
You may object that this is all very well when the person is interesting, and that a good deal of what people say isn't. Your father-in-law is telling you about his route down the A34 again. I'd make two points. The first is that interest partly follows attention. When you half-listen, you hear the general outline, and the outline of the A34 story is dull. If you do listen, there's sometimes something in the particulars. It may be why he tells it, or what he's proud of. Sometimes you'll notice that he's lonely and this is the conversation he knows how to have. A follow-up question can take a stale story somewhere new. The second point is that you don't have to do this with everyone all the time. Chapter 14 suggests starting with one conversation a day. Choose the people who matter to you, and let the A34 take its chances.
Your own contributions still get made. Once you've made your first response and the person has said what they needed to, your turn comes round, and it's a better turn. The advice you give now takes account of the whole story. If you tell a story of your own, it's one that their conversation called up and not one that pushed theirs aside. You'll find that replies composed in the second after someone finishes are generally better than the ones composed during their third sentence, because they're built from more. They also come quickly. The blank mind you fear lasts about a second, if it comes at all.
There's one more thing to say about the fear of seeming slow. People don't judge how good a conversation was by how quickly you came back or by how clever your reply was. They go by how they felt afterwards. If someone listened to the end and asked one good question, they'll remember you as excellent company, even if you said very little.