In what way do you want the girl you're looking for to be 'better' than your best friend? Because it sounds a bit like you don't know, and are just looking for someone exactly like her, which isn't a good course of action because you're unlikely to find someone like that. Standards are good, but not if they're so restrictive that they eliminate everyone. As has been said already here, the people who you find yourself liking are usually the people who you never thought you would. And also, if you're still pining for your friend, then maybe it's not a good idea to look for someone new, because you'll risk hurting them.
So yes, don't throw your morals out the window. Having moral standards is a good thing. And don't actively seek a new partner if you're still hung up on your friend. If you meet someone you like, then hurrah, but looking for someone like her will only lead to badness (and not in a good way).