No. You already filed a return. You should not e-file it now or you will cause confusion and delays with the IRS. The IRS does not want two tax returns from you. You need to wait for the IRS to process the return that you mailed.
TurboTax does not receive any updates from the IRS. Your ONLY source of information about your refund now is the IRS.
You need your filing status, your Social Security number and the exact amount (line 35a of your 2023 Form 1040) of your federal refund to track your Federal refund:
To track your state refund:
https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund
https://ttlc.intuit.com/questions/1901548-why-do-some-refunds-take-longer-than-others
https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions
You do not want to efile a return after it was mailed. This will cause issues and delay processing of your return.
No, you should not mail a second return to the IRS at this time.
The information in the link below shows that the IRS is currently processing mailed returns received in March. You should try to contact the IRS again for some further advice using the 1-800-829-1040 phone number.
Processing status for tax forms
If your proof of mailing also contains tracking information, try to verify that the package was delivered. If you find that it was not delivered, then you should mail your return again.