Yes, if you bought it from Amazon it'll be the full game. I didn't know that Amazon sold the Steam version, but if that's how you launch your game then I assume that you'd activate via Steam. Try right click the game in your Steam games list and see if there's an option there somewhere to enter the key? (I assume they'd send your key via email, or within the box if you got a boxed version)
If you can't find any option there, then I'm not sure how you'd activate.
Just to verify that it is a problem with being in trial mode, and not some other issue - have you played single player yet? And if so, have you been able to go beyond level...8, I think it is? If you're in trial mode, after you reach that level you'll get a message saying that your game has been saved, and you'll need to activate it to continue playing and leveling that character.
If you've leveled much higher than that, and gotten no such message, then it seems that your game is activated and that it's some other issue that's giving you an error with your authentication code in multiplayer. If that's the case, I'd search the tech support forums and see if you can find some way to resolve the issue. If you have no luck with a search, start a thread there and see if someone can help you with the problem.
Edit: OK apparently the thread was moved to tech support already, so ignore my advice to start a new thread if a search doesn't turn up anything.