If a bank accepts a post dated check and it causes your account to become overdrawn, is the bank responsible for the overdrafts or am I?

No, the bank is not responsible. You are responsible. Most banks do not look at the date, the check is legal tender, and it can be cashed when received. If you read the Uniform Commercial Code, a bank will take a check regardless of the date. So, if you've postdated a check, and the payee cashes the check, that's the date the funds will be taken from your account - - or if there are insufficient funds, then you'll be in hot soup. And no, it's no longer a defense to the principle amount and the INF fees to say, "Well, it was postdated!"

See the following threads:
https://www.askmehelpdesk.com/banking/post-dating-checks-30207.html

You are.

never write checks unless the money is in the bank to cover it. And don't send post dated checks to creditors.