Don't do it. Wait for the W-2. If you submit your tax return and it does not match what your W-2 shows, that could cause months of delays. If you are unable to retrieve your W-2, the IRS will allow you to file a substitute W-2, but in those rare cases, you cannot e-file, you have to file by mail.
If your return gets accepted, but doesn't match the IRS database (and it likely won't if you are using your last paystub), your return will go to manual processing and any refund due will be delayed and considering how far behind they are, it will probably be delayed for a long time. It is better to wait a week now than to wait months for your refund later.
Wait for your W-2 and let TurboTax correctly determine your state and local tax situation.