Yes, do have your membership card available and ready to show. I have been asked to show it in the past at check-in and it does have an expiration date on it.
Also, having a valid AAA card is good, not just for Disney discounts. I ended up with a flat tire the other day, with my husbands car and he was out of town. I called AAA, had it changed in a heartbeat with no out of pocket expense to me other than my yearly AAA annual fee. And, that fee was paid for previously due to my Disney hotel savings this past year.