If you are not able to electronically file your amended federal return, then yes, you should mail it in. I recommend using certified mail receipt so you have proof of delivery. You do not need to wait until your federal amended return is processed to amend your Illinois return.
As for your Illinois return, if you already filed your original return, you aren't considered late in filing and wouldn't need a state extension. You would simply be amending (correcting) an already filed tax return.
Please see this link from the IRS to track your amended return.
@mshaikh010
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