Hello and :welcome to EBN!
yes, teething and runny eyes go together. As does the drinking too much water and throwing up,
my girl is 8 yrs old and still eats/drinks this way. For food, you can try the slow feed stainless ball -- you can get them on Amazon. I add a little water to Banks food which does slow her a little ... Water I put ice cubes and she drinks around them.