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There's nothing wrong with cancelling cards but in your situation it sounds like it would be a bad idea. You are already in with 3 very good banks and want to cancel two of them. As others have noted, the APR doesn't matter if you aren't carrying a balance. I would not end relationships with Chase and AmEx when your profile is so thin.
Good advice here, OP. Very thoughtful and purposeful. I hope you decide to follow their recommendations. Best of luck to you.