The best lotion I have found is VaniCream from the pharmacy. It is kept behind the counter... they use this to mix many medications with. It contains NO harmful chemicals and is extremely gentle on your skin and will add the moisture you need without being greasy or heavy feeling. Go to Vanicream.com and you can read more about it before you go to your pharmacy to get. It also is not that expensive and a little goes a long way!
Stop with the hand sanitizers unless you are only using them on the palm of your hands and fingers. Do not put on the top- it will just continue to dry the skin out! Also, try to change to a regular soap or a mild anti-bacterial one- I use Bath and Body Works foaming one and it helps... the one that looks creamy- forget the name though works great although the last inch in the bottle never comes out so a waste of money. ALso, I am really cheap, I put a rubber band on the pump neck so I only get 1/2 pumps since it is a waste to use more!
When drying your hands... do not dry completely before putting lotion on. Let the lotion trap the moisture in them. STOP with the peroxide (remember what it did to your hair as a teen lol). Neosporin is good... I use A&D ointment lightly on the tops of my hands at night right now and that helps too.
When going out in cold weather make sure you are wearing gloves. This weather will rip up the best and toughest of skin!
Good luck! I know how painful this is- I have OCD and am a hand washer! I am trying to work through the dry bleeding hands myself and those are the things that work best for me!
Blessings to you and yours!