I'm not in Massachusetts so I don't know if they're sending "accepted" information back to TurboTax. However, if the DOR website says it was received on or after the date & time you transmitted it, that is more definite than TurboTax. (If they received it *before* you transmitted it, contact DOR to make sure you're not a refund fraud victim.)
As far as refund issuance, I do know that many states are taking much longer with refunds this year in an effort to catch refund fraud. (My own state, Arkansas, added a message to the results on its Where's My Refund site warning taxpayers of this delay.) Generally, you should always allow 3 weeks for any refund from an e-filed return, longer for Federal returns claiming EITC or ACTC (which won't be released till mid- to late February).