Not sure what IRS site you are looking at to see if your return was processed. Are you looking at your IRS online account?
https://www.irs.gov/payments/your-online-account
And...you mention "a check"---did you mail your state return to the state with a check made payable to the state? The IRS has nothing to do with your state tax return. The IRS processes your federal return; the state processes your state return.
You say that your check(s?) was cashed, so you need to hang on the the cancelled check as proof that you paid your tax due.
And..no, the IRS does not send you any paperwork or notification to tell you that they processed your return. If you filed and paid before the April 15 deadline you should be fine.
**Disclaimer: Every effort has been made to offer the most correct information possible. The poster disclaims any legal responsibility for the accuracy of the information that is contained in this post.**
