Is not such a thing as hair dye for normal, dry or oily hair. The only thing you can do is to use the right shampoo and conditioner for oily hair. You should always colour your hair when it’s oily, the natural oil provides some protection for your hair. I recommend you to have your hair dye always made by a professional hairdresser as the high street brands can sometimes make it worst for your hair.
On average there are 100,000 hairs on the human head. Each hair grows for about three years before it falls out and a new one grows. Around 50 to 100 hairs fall out every day, no matter the length of the hair, so it is not necessarily a worrying sign to see hairs falling out. If you realise the hair loos is much more then usual , I recommend you to see your GP first for some blood test as this can have other causes as stress, hormone imbalance, Iron deficiency etc.
Also styling products and heating equipment , if used in excess, can make your hair weak and easy breakable so you’ll say your hair is falling when actually is braking I recommend you to trim your hair every 6-8 weeks and use hydrating hair masks 1/week . If you have damage hair ends then I recommend you to give them a trim asap and see if you realise an improvement in your hair loss. When your hair is long you realise more the normal daily hair loss then when is short.