So I tried to start the quest with Sam in it and it wouldn't let me. I found him at the Inn in Markarth and did the drinking contest but for some reason my character won't black out. He drinks all the drinks and then just walks away. I've tried at least 10 times and still nothing. Help?

Try reloading from before the time you entered Markarth?

Seems there is a bug similar to the one you described. http://elderscrolls.wikia.com/wiki/A_Night_to_Remember

Thanks, found out it's because I killed Ysolda when I was overthrowing Whiterun.

Great, you found the only non-essential NPC in Skyrim and, as it turns out- she's essential