I don't think you have anything to worry about. As long as you have an ESTA that is authorised that is all that you need. You don't even need a copy of it as it is held electronically.
Your application wouldn't have been accepted if all the relevant information was not input at the time so just because you can't see it on your print out doesn't mean that it isn't on the system.
Just for info, the ESTA only needs to be valid for your entry to the US. It doesn't matter if it expires while you are there. It is an Electronic System for Travel Authorization, not a stay authorisation.