Well, excuse my language, but the basic technique is called The Shit Sandwich.
This is the same technique used by some HR professionals to deliver negative performance reviews - they turn it into positive-negative-positive. For example: 'Your time-keeping is improved, but your overall performance is still well below your targets. I know you can do better than this.' Prospective authors will also recognise this formula from countless publisher's rejection letters.
Obviously, in order to be effective, the three elements (good, bad, good) need to be logically linked. The classic bad example is 'it's a lovely day today; you've got bad breath; I like your shoes'.
So, basically, you should deliver first good news, then the bad news, then good news. If you're saying 'no', then you can and should substitute empathy for good-news according to context.
Example 1: "That sounds fun" --- "But, no, I'm afraid I just can't come tonight" --- "I hope it goes well"
Example 2: "You poor thing" --- "I'm already snowed under, so it's a no" --- "I'm sure it'll be fine"
Example 3: "How exciting" --- "No, I can't do anything this time" ---"Let me know when the next one is"
[Note that even though these examples aim to sweeten the pill, they still include the magic word 'No'. This is important, because there are powerful counter-techniques used in persuasion, appealing to your good nature, your unique talents, your moral principles etc. You need to be prepared (literally prepared, through rehearsal) to say the word: NO ]
Now let's be clear about this: the actual formula itself really isn't that important, in fact it is, as they say, pretty naff. The magic appears when you actually start thinking beforehand about how you are going to say no, and mentally prepare yourself for that eventuality as a routine part of your working life. In other words, you need to get used to it, and you need to be prepared for it, treating it as just another task that needs to be planned beforehand ('it' being 'no').
This simple insight seems to be ignored by many in education: the notion that 'saying no' is simply another part of a teachers skillset, which needs to be understood, practised and developed, along with differentation, behaviour management, lesson observations, and a host of other everyday teaching facets. If you look at it in this way, it should really be a part of teacher training, shouldn't it? Teachers are always trained how to deal with difficult students, usually taught how to respond to difficult parents, but almost never trained how to deal with difficult colleagues or 'superiors'.
If you are well-versed in TM4T, you will doubtless snort at this: "It's just the same old respond-don't-react mantra yet again, isn't it?". Well, yes.
You may also want to explore other, complementary techniques alonside the shit sandwich:
a) Request more information: 'how long would that take?', 'how often mightyou need me?', 'what exactly does it involve?'. Note the vocabulary here involves conditional words like 'would' and 'might', not 'will'. However, make sure that you do reply (whether it is 'yes' or 'no').
b) Defer: 'I'll see what else I've got on and let you know'. Note that this is not 'delay' for its own sake. You are making sure that any decision you make is logical rather than emotional. Again, make sure that you do reply conscientiously. Never just fob someone off in school, it might work but it is counter-productive in the longer term.
c) Negotiate: 'If you take my Year 9s on Tuesday, I might be able to juggle things and fit it in'. Again, make any offers conditional, starting 'if' and ending 'might'. You may have read about the old politician's trick (and parlour game) which involves never saying the word 'yes'. This applies during negotiation.
You may also consider giving a little thought to body language. Those working in customer service (flight attendants are the classic example) are often trained to supplement a negative verbal message with positive body language: sometimes a sympathetic smile and an open posture can remove the harshness from a 'no'.