I went for a fill today and talked at length about this with the nurse.
She said that at one time it was thought that drinking while eating could cause issues with the band. However research has shown that it is not as important as it was first thought. She was saying that its perfectly ok to drink up to the time of eating (and as such if you're feeling hungry but not sure if its hunger it can also help to see if it genuinely is or not) she said they still however say to drink as little as possible while eating, although small sips would be ok, but having liquid with food will flush the food through the stomach more quickly and this is something that you want/need to avoid to sustain the feeling of fullness. She said there's no real need to delay after eating, but you may be too full to drink and again, small sips would suffice until the food has passed through to resume normal drinking then.
My provider even removed the info from their book inline with this too.