Use awkward pauses as a tool to say no. When a request comes to you (this works only in person), just pause for a moment. Count to three before delivering your verdict.
E-mail is also a good way to start practicing saying "no but" because it gives you the chance to draft and redraft your "no" to make it as graceful as possible. Plus, many people find that the distance of e-mail reduces the fear of awkwardness.
It gives you the time to pause and reflect and ultimately reply that if you are or are not available. It enables you to take back control of your own decisions rather than be rushed into a “yes” when you are asked.
Using email bouncebacks is the most socially acceptable “no” there is. It doesn't mean you're telling them that you don't want to respond to them. It only states that you can't reply for a certain period of time.
Saying no to a senior leader at work is almost unthinkable, even laughable, for many people. However, when saying yes is going to compromise your ability to make the highest level of contribution to your work, it is also your obligation. In this case it is not only reasonable to say no, it is essential. One effective way to do that is to remind your superiors what you would be neglecting if you said yes and force them to grapple with the trade-off.
Saying "no" with humor will lessen the awkwardness and will lighten up the atmosphere.
This is a particularly good way to navigate a request you would like to support somewhat but cannot throw your full weight behind. I particularly like this construct because it also expresses a respect for the other person's ability to choose, as well as your own. It reminds both parties of the choices they have.
It is tempting to think that our help is uniquely invaluable, but often people requesting something don't really care if we're the ones who help them- as long as they get the help.