Yes, you need to mail them in. Massachusetts needs proof of what you are claiming in Rhode Island to process the credit you claim for taxes paid to Rhode Island.
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No, the MA NR/PY instructions do not say anything about including a copy of other state returns with your MA income tax return.
TurboTax is just printing all your state tax returns. In any event, you should _always_ make a printed or PDF copy of your returns even if you e-file. Just in case!