As far as I know now, it's not possible to have the signature change when you click on the account button to change the account?
You can assign a signature to a account in the panel opened at tools/options/mail format tab. Don't click the Signature button unless you want to Edit, create, or remove a signature. Of course, you must have multiple signatures before you can selectively assign a signature to an account. At tools/options/mail format tab, you should see a drop down arrow for "Select signatures for account:". Select the account in that box. Then, select the signature you want to assign to it in the box below that one. Click Apply, and OK.