Yes that is correct. The fee was not for e-filing. The fee was for using the software to prepare the return instead of having to sit there with a calculator and paper forms. The fee is the same whether you e-file or mail it.
You will receive an email and if you signed up for text messages, you will receive that once your returns are accepted or denied.
Please see this Turbo Tax FAQ on how to track your State refund. Track state refund
You can check to see if your return was e-filed using this tool.
How do I check my e-file status? E-file Tool
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