Can I open another bank account at a different bank if I owe overdraft charges at another?

Banks report overdrawn accounts. Pending on the bank and if they credit check you may or may not be able to.

There is no law that you can not, but normally it is reported to the agency banks check for over drafts and other bank fees.

So normally they will check, find the problem and not allow it,

But if you try enough banks you may find one.

The issue is, you still owe that money,