I have already filed my tax returns. I owe state, but I can't pay it by April 15th. Can I get an extension?

An extension does not give you extra time to pay.  It only gives you extra time to get your documents in order.  You are still supposed to pay by the tax deadline.

Some states allow installment plans if you are not able to pay everything up front.  This FAQ has more details: https://ttlc.intuit.com/questions/2840416-can-i-set-up-a-payment-plan-for-my-taxes