Oh, lady! First of all, this situation sucks. Take heart, horrible complaints happen to us all... Ugh, the stories I could tell!
Don't beat yourself up, and (omg I sound like my dad!) this was a great learning opportunity.
You don't know the personality of your customers and they don't know you. At the first, slightest, ever-so-subtle, hint-of-a-complaint, misunderstanding, grumpiness, whatever, I send a neutral party over to do damage control. Manager, hostess, another waitress...
Sometimes it's just to chat them up, other times to help soften a mistake (like, "Hey, Christine is concerned we are are not doing all we could be to make sure you're happy. Is there anything I can do to change that? Tell me what needs to be done, and I'd love to do it!"). It shows that you are genuinely concerned enough to get someone else involved.
(Hopefully this doesn't come across as preachy! Just wanted to offer the point of view of someone who has done this for a million years... I've been a waitress, a patron, and a manager. This tactic has been successful for me no matter which shoes I was wearing.)