Estimated Payment to State

Do NOT pay twice ... you either mail in a payment with the voucher   OR   do it online not BOTH unless you like paying twice. 

I mean making the payment online, taking the print of online payment proof and mailing them with form  2018 Form M-4868 Massachusetts Extension  Payment Voucher 

NO ... do not mail anything if you pay online ... they know you paid ... they do not need proof.  And it is way too late to file an extension for 2018 or 2019.    If your 2019 extension payment did not  go thru with the extension it is too late  to pay or mail it in ... you only had until midnight on 7/20/20 to "perfect" the rejected extension and payment so now it is just late. 

Can I pay the estimated tax liability now online?  so that i do not get penalized further 

Of course. 

and I was because last year I paid some amount with extension to state and today I called them and said they have not got any payment I have an email from them for payment proof and the payment appears on my bank statement. 

Since the 2018 return MUST be printed and mailed  in   attach the  proof of the payment made along with all the other forms you need to attach.