If I amended my federal return and added two dependents, do I need to send in an amended state return too? or just federal?

Yes you must also amend your state return and mail it to the state.

I amended my 2018 return today March 7th, 2020, because my second child (dependent) was omitted from the original tax refund application.

The issue now is that I had mistakenly mailed the 1040X with 1040, EIC. ACTC forms to the IRS.  Because when I printed it on TurboTax, it printed the 1040X with 1040, ACTC and EIC. So, I mailed them all to the IRS for refund.

What is the way forward?

Should I resend the mail by sending only 1040X alone? I can see the 1040X is just 2 pages.