If I was sued by a debt buyer who purchased old credit card debt, what can happen to my savings and checking accounts?

If your creditor is successful in obtaining a judgement in court, then all of your "non-exempt" assets are at risk. In your situation, this appears to be your bank accounts. This is true whether the accounts are solely in your name or if they are joint accounts (your name plus someone's name). So any income you receive should not be put in your name.

That having been said, unemployment benefits are "non-exempt". This means that they cannot be seized by creditors. At least in theory. What could happen however, is that your bank account is "frozen" by a creditor and then you would have to demonstrate that the funds in the account were from unemployment. At that point the funds would be released. If, however, you don't do this within the specified time allowed, even though the funds are otherwise exempt, they can be released and paid over to your creditor. Again, the better course of action would not be to put those funds into a bank account with your name on it.

You should be aware, that if your creditor sues you in court, you must be notified of the proceedings beforehand. You could then close your accounts at that time. However, you should be further aware that certain unscrupulous collection agents fail to give the proper legally required notice and obtain a judgment without the debtor present in court. If that's the case, you can fight such a judgment. However, in the meantime your assets are at risk. So once again, keep them out of your name to play it safe.