First, dry scalp does not lead to hair falling out unless it is really severe but it will change the consistency of hair though making it more brittle. Second, dry scalp can be uncomfortable though, as it can be accompanied by itching if let go, and to flaking which are signs of more adavance dry scalp, but there are a lot of products that can help with it to combat it. No worries.

Shampoo and Conditioners: Head and Shoulders, Neutrogena T Gel, Nizoral are some of the top three I can think of that are in the grocery stores. The first two are better for mild to moderate cases of sry scalp, itching or flaking and if you have color on your hair, as it won't strip it, these are also the least expensive; Nizoral used to be only available as a prescription, is stronger that the others in that it will strip the color from dyed hair, but is good for severe cases, with severe flaking and itching too - It does not sound like you are to that point (?). More/stronger is not better when it comes to treating your scalp so going with the appropriate level is important. I would start out with one of the first two or Nioxin next...

Then there is Nioxin ( cleanser and scalp therapy conditioner) is at the beauty supply stores or salons, it works really well too but is more expensive, and will not strip color from the hair. The beauty supply store also has treatment packets of a power that can be mixed with water to make a paste that can be applied to the scalp to help as well, but these are messy yet really good at what they do (there are different brands, ask an associate).

Unfortunately it is trial and error to see which works best for you. I hope you find some help soon and the product that works best for you.