You're just really needing to moisturise under your eyes with eye cream, or a moisturiser that can be used on your eye area. Try moisturises with vitamin E or Aloe vera in them. Many vitamins like vitamin E help sooth and smooth your skin.
A good cleansing routine, where you apply moisturising products ( like face wash, and gentle exfoluents) to your face to keep your skin healthier from the inside and outside. As you know, water is great for your skin, but just that alone won't keep it smooth, that's why it's important to moisturise.
Also having the right cleansing tool will make things so much better, other than using your hands, fingers, or a cloth, use an electric sonic brush, it will clean your face 6x better and will get rid of dead skin cells, dirt, and impurities more effectively.
I also suggest, having smooth skin depends on water temprature, even though warm water is great for opening poors with steam, it can dry out your skin if you only use warm water. Cold water closes poors, cools down your face, and locks in moisture, so it's better to use less warm water. Only use it to open your pores before washing your face, and to wash off the product, switch to cold water quickly to lock in that miosture.
Depending on your skin type, are you likely to get really oily, slightly with dry patches, or very dry? (you mentioned those dry parts so I assume you have dry skin or slightly oily with dry patches, otherwise known as combination.
Hopefully I have helped with my answer. :)