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If the cards aren't costing you anything, keeping them won't hurt you. Closing them out can hurt by lowering your available credit and raising UTI. It will not hurt your AAOA as it will still be reported for however long you had them opened for.
Even if you close the cards, reports will still show the limit on the cards. All of those Cap One's and the BoA may be bucketed and no room for CLI's. Those may just need to be throw in the sock drawer. Focus on both Amex and the Discover IT as these are the cards that most likely will be more generous with CLIs. When did you get these 3 cards?
Do you have any derogs? Whatever is holding Amex back from giving you any CLI's is probably going to hold you back from getting a higher SL with a new card.
Amex is known to be very hesitant to open up until one's file is clean. I'd wait until those accounts fall off and score changes are reflected in your Amex account before trying again with them. Having some old lates may be ok since they're years back, although I can't promise it, but if you have CO/CA accounts you're stuck until they're gone at least.
I would wait until those derogs falls off. Getting that UTI lower would also help too.