They shouldn't do....I use a small chick feeder and drinker. As long as they cannot get themselves stuck in it in any way, they can't drown.
Your chicks don't NEED food or water for about the first 72 hors BUT I always give food and water as soon as they are out of the incy and into the brooder and nine times out of ten, they show some level of interest in it within the first 24 to 48 hours. As long as your chicks are alert and perky looking when they get up and walk round, that's fine. As soon as they lay down they tend to go very sleepy and droopy, often falling asleep face down or flat out, looking very poorly but actually being just fine. It is their awake behaviour you need to look at, not their dozing. They should look sprightly, peck at things around them, be bright eyed and curious. If you see any lethargy or lack of enthusiasm, a drop of Nurtidrench or some scrambled eggs will give them a boost. It is not at all unusual for them to be very tired out for the first 12 to 24 hours after hatch though, especially if there have been any problems or delays.....as there has been with your second little one.
Watch out for pasty butt when they start eating....this is where their droppings stick to their bum fluff and dry out, causing the vent to become blocked and unable to pass further droppings. it can kill quickly and yet it is so easy to cure.... wash the droppings off with warm water, dry the area thoroughly, add a small drop of olive oil to stop further adhesion and add a teaspoon of ACV to their water drinker. The scrambled egg is good for drying up pasty butt too.