I forgot to sign my amended return before submitting it. What should I do?

Wait for the IRS to act.  They will either process it or send it back.  (The odds are greater that they will process it if it shows you owed money and sent a check.  If it's a refund claim, they almost always send it back for signature.)

I forgot to sign my mailed state tax return for Massachusetts. What happens next?

What if I signed the return? But forgot to sign the check? Any chance they will provide me a way to pay without me remailing all the stuff all over again?

I think your bank will pay the IRS if IRS submits the check with no signature.

otherwise, you will be hearing from IRS.