My husband and I file state taxes separately on the same return. Does he still need to file taxes on his own, or is it covered because we're filing on the same return?

If his income is shown on the married filing separate return in a column and yours in another column, your state return has been filed and nothing else needs to be done.  

If this is NOT the case and his income is not in a separate column on same State return, then he will need to file a separate married filing separate return. 

There are some states where you can file Married Filing Joint or Married Filing Separate on the same tax form.  I assume yours is one of these states (as I do not have your information, nor should I).