If the account is paid off, you should be fine. PLUS Loan credit checks
mostly look for default on any student loan debt and don't take credit scores into account. Should your PLUS application be denied for credit reasons, your student would become eligible for an increased amount of Stafford loans.
PLUS Loans are only in the name of the parent. If you'd rather cosign a loan that's in the student's name, you should look into private student loan options