How do I know if the incorrect tax return was processed by the IRS already?
Processing paper returns is usually 4-8 weeks -- unless you filed with ITIN application (W-7) in which case it's a good bit longer but I don't know how much currently.
You could see if you qualify for their online get-transcript application -- right up front it says it only works for SSN holders, and although it doesn't say explicitly they recently had to make the authentication more difficult so that even 'normal people' (citizens, in the US' perspective) have trouble passing (according to the Taxpayer Advocate) because previously it was repeatedly hacked and now "we can't have nice things". If not, wait until after the filing crush ends, maybe April 24 or so, and periodically phone the accounts line (800-829-1040) to check.
How do I correct for this mistake?
Once the original return is processed, file an amended return using 1040X plus a corrected 1040NR. See F1 student in US, correcting mistake in wrong tax form filed for a similar but more extreme case (but I endorse the approach of letting IRS compute the interest, doing it yourself is a serious pain -- and for a month or two it won't be much anyway).
Is it possible to file my state tax returns using the correct 1040NR even if I filed an incorrect one for my federal? (I have to file both MA and CA tax returns)
Absolutely. Some states (I don't know MA and CA specifically) do data matching with IRS to catch cheats and/or mistakes, but only well after filing season is over, probably August at the earliest. You should be able to get your correction in by then. Worst case, if the state does catch this and send you a query, just send them a letter explaining why the figures you filed with them were correct.