The words
Once you're at the kitchen table and the answer is no, you have to write it. People make this harder than it is. They draft four paragraphs, or they wait three days for the right wording to come, and by then they're back in the territory of chapter 11.
A good no is short and has two parts, with a third if you choose. You say something warm that's also true. Then you say the no, in words that can't be mistaken for a maybe. The optional third part is a smaller thing you'd be glad to do.
Here are three, as examples.
"Thanks for thinking of me for the tombola. I can't do it this year, I need that Saturday. I hope it goes really well."
"I've had a look and I can't take on the treasurer job. It needs someone with more evenings free than I've got. I'm happy to show whoever does it how the spreadsheet works."
"That's kind of you to ask. I'm going to say no to the weekend away, but I'd love to see you both. Could you do a Sunday lunch next month?"
None of the three apologises more than once, and the first and third don't apologise at all. None gives a reason that can be argued with or fixed. "I need that Saturday" is a fact about you. Nobody can offer to move it. And in each one the no is a plain statement, "I can't" or "I'm going to say no". It isn't "I'm not sure I'll be able to", which the other person will read, hopefully and reasonably, as a possible yes.
I'd leave out the following things.
Leave out the long apology. One "sorry" is fine if you mean it. Five of them tell the other person you think you've done something wrong, and they may start to agree.
Leave out the invented reason, for the reasons in chapter 11.
Leave out "maybe next time", unless you'd welcome being asked next time. If you wouldn't, you've booked yourself another of these conversations.
And leave out the request for permission. "Would that be all right?" and "I hope that's OK?" hand the decision back to them. You're telling them your answer. You aren't applying for it.
The offer in the third part deserves some care, because it can smuggle the old habit back in. "I can't do the whole day but I could do the morning" is fine if you'd tick yes to the morning on the slip of paper. If you wouldn't, you've given an unwilling yes to half the job. The test for the offer is the same as the test for everything else. When in doubt, leave it off. A no with good wishes is complete on its own.
Sometimes the other person comes back. "Are you sure? It's only a few hours. We're really stuck." Say the same thing again, a little shorter. "I know, and I'm sorry you're stuck. I can't do it." Don't add a new reason. Each new reason is a fresh thing for them to answer, and it suggests the first reason wasn't the real one. People who keep getting the same friendly sentence run out of replies quickly. It almost never goes past the second round.
When the day comes that you'd like to say it to a face, the words are the same. "Thanks for asking. I can't, I'm afraid, but I hope it goes well." Say it at normal speed, and then stop talking, which is the difficult part. The silence afterwards lasts about two seconds and feels like ten. The other person nearly always fills it with "no worries" or "fair enough". The moment you've been paying whole Saturdays to avoid has then come and gone in the time it takes to put a lid on a bin.