What limb does he desire?
Radak: He desires a wing like a dove, so he could fly to a place where they cannot catch him. Then, he would dwell without fear.
Why does he mention "Yonah"?
Radak citing Ibn Ezra: It grows among people, and kings use it to send their letters.
Radak (from Bereishis Rabah 39): All birds, when they get tired, they rest on a rock or tree, and they are caught. When a Yonah gets tired, it closes (withdraws) one wing and flies with the other, and it is not caught.
Malbim: David decided to flee Yerushalayim, but he did not yet flee. The people passed, but David stayed. He wishes that he had a wing like a dove, for it can fly with one wing and rest with the other. He wishes to flee and not flee, stay still and not stay still; he was indecisive.