No corpses so far. Wights in the books are described as having pale flesh, pale blue eyes, black hands. They greatly differ from classic Romero type zombies.
A headless wight or wight's severed arm still poses a threat. So there are no other way to kill the wight, except the fire.
ADWD, chapter 13, Bran. This chapter is a good source, it has all the answers for your questions.
All around him, wights were rising from beneath the snow. Two, three,
four. Bran lost count. They surged up violently amidst sudden clouds
of snow. Some wore black cloaks, some ragged skins, some nothing.
All of them had pale flesh and black hands. Their eyes glowed like pale blue starts.
Bran filled a fist with snow and threw it, but the wight did bot so
much as blink. A black hand fumbled at his face, another at his belly.
Its fingers felt like iron. He's going to pull my guts out. But
suddenly Summer was between them. Bran glimpsed skin tear like cheap
cloth, heard the splintering of bone. He saw a hand and wrist rip
loose, pale fingers wriggling, the sleeve faded black roughspun.
Black, he thought, he's wearing black, he was one of the Watch. Summer
flung the arm aside, twisted, and sank his teeth into the dead man;s
neck under the chin. When the big grey wolf wrenched free, he took
most of the creature's throat out in an explosion of pale rotten meat.
The severed hand was still moving. Bran rolled away from it
"Hoooodor" came whimper, from somewhere down below. And suddenly he
was not Bran, the broken boy crawling through the snow, suddenly he
was Hodor halfway down the hill, with the wight raking at his eyes.
Roaring, he came lurching to his feet, throwing the thing violently
aside. It went to one knee, began to rise again. Bran ripped Hodor's
longsword from his belt. Deep inside he could hear poor Hodor
whimpering still, but outside he was seven feet of fury with old iron
in his hand. He raised the sword and brought it down upon the dead
man, grunting as the blade sheared through wet wool and rusted mail
and rotted leather, biting deep into the bones and flesh beneath.
"Hodor!" he bellowed, and slashed again. This time he took the
wight's head off at the nek, and for a half moment he exulted... until
a pair of dead hands came groping blindly for his throat.