If I owe state and efiled, do I need to include anything with my check I mail to State Dept of Revenue

what state? most have a form like the IRS 1040-V to accompany the payment.  virtually every state allows payment through its website thus avoiding the need to mail in the payment and thus avoiding the possibility that the check gets lost by the USPS. 

It depends on the state.  For most states you should have a voucher that prints out in a coupon format similar to the IRS' voucher.  You would clip off the bottom portion and include that with your payment.  Don't forget to follow the instructions provided on how to make out the check, and if you need any particular wording in the remitter line such as your Social Security Number and/or the tax year/tax form involved.