I say invest the whole inheritance in an account you will not touch. Lock it away for a very long time! Keep paying off you credit cards as you had planned without the inheritance money. Yes, it will cost you more in interest than paying them off now, but I would still do it that way. A large lump sum of money is VERY hard to build and so easy to spend. Most people will make sure they pay their bills, but often, even with the best intentions, will not ever completely pay themselves back (or save money) because things always come up. The mindset often becomes if you don't pay the cc company, you get in trouble, if you don't pay yourself, no big deal.
I'm with you. From a strictly numbers point of view, the smartest financial move would be to pay off the credit card debt. However, I interpret OP as saying that if she pays off the credit cards, she'll quickly run them up again spending on anything and everything. In this case, I think it's better to put the money aside and chip away at the credit cards normally. And if it's too hard to resist using them, close them when they're paid off.