TurboTax said that the IRS had approved and was about to deposit my funds by the 27th. When going to the IRS website, they say they haven’t even approved my tax return??

The TurboTax dates are only estimates.

 

Only the IRS and your State control when and if a Federal or State tax refund is Approved and Issued.

 

Once a tax return has been Accepted by the IRS or a State, TurboTax receives no further information concerning the tax return or the status of any tax refund.

 

If accepted by the IRS use the federal tax refund website to check the refund status - https://www.irs.gov/refunds

If accepted by the state use this TurboTax support FAQ to check the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund

 

After the tax return has been Accepted by the IRS (meaning only that they received the return) it will be in the Processing mode until the tax refund has been Approved and then an Issue Date will be available on the IRS website.
See this IRS website for federal tax refund FAQ's - https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions

TurboTax does not tell you when a tax return is approved by the IRS. TurboTax will tell you whether it is "Accepted" or "Rejected" but not when it is approved.