Yes, CX crews bring big suitcases that they check in at the counter -- I've seen them do it! I always assumed it was by choice, since they wanted to shop in the States during their layover.

As far as AA, I typically see AA FAs bringing a rollaboard, personal item, and small cooler with food for the day's flights. Maybe a purse as well. Typically a total of 3-4 items. Remember that while pilots still get crew meals per their contract, FAs do NOT. So, FAs have to scrounge what they can from leftover premium cabin catering (NOT the BOB items, since those items are inventory-controlled). OR they have to bring their own food from home, in order to avoid overpriced airport food.

I'm not an FA, but methinks the life of an FA working domestic/Mex/Can/Carib routes is not as glamorous as it used to be. Only the longhaul routes still have some pretty cool lifestyle perks (weekends in Paris or Buenos Aires, enjoying gelato in Rome, etc), but the seniority required to fly bid for those routes is off the charts these days.