Generally speaking, during a fermata, as long as you have the pedal down, you can do whatever. However, if you just do whatever, it means you're not listening anymore. And somehow (psychologically, magically, physically, whatever I don't know) listening goes an incredibly long way in adding to your performance. Actually, as you listen to your sound, your body makes minute adjustments so that you get the sound that you want, instead of just playing keys.
However, since we're talking about something where you hold down the pedal and whatever, listening seems a bit more mystical. But it makes a huge difference, and that's why most of us here don't feel awkward during fermatas. Just listen to the sound, whether it's a chord, single note, or silence. Feel the intensity of the moment. If it's a note, hold it with your fingers even if it doesn't change the sound. Psychologically, you're more likely to listen to a note you hold with your fingers. If it's silence, don't make unnecessary movements (like resting your head). It's only awkward if you think it's awkward and do awkward things.
Although there is a point where you can just hold the pedal and do whatever, the music should never feel whatever. Never. It's an insult to whoever's listening.
On a related issue, preparation. After a fermata, if you have something else coming up, don't prepare early. You lose the impact. Do prepare so you don't miss it, but at the last possible moment.
Btw, if you're fermatas are long enough to feel awkward about, they're probably too long. Just an idea.
Topic: What do you do on a fermata?