No, do not use the Non-Filers tool. The "Way to go, you filed your 2019 taxes" just means that you finished your return with TurboTax. Since you had to mail your return TurboTax will not get an acceptance notification. Did you send the return with tracking? Then you can know that the return got there.
"The IRS is not currently able to process individual paper tax returns. If you already have filed via paper but it has not yet been processed, do not file a second tax return or write to the IRS to inquire about the status of your return or your economic impact payment. Paper returns will be processed once processing centers are able to reopen". (IRS News Release)
About the Stimulus payment, if you filed a 2018 return then you should still get a payment if you meet the requirements even if your 2019 is not processed yet.
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