Gross nails! What do I do?

My cuticles have been growing so long like really really long and then peel. I also get a lot of excess skin around my nail that peels like crazy. Maybe from all the sanitizer and hand washing but I try to use lotion too and they still do this. Sometimes the skin gets hard a d it peels off and I have big missing skin chunks around my finger and it looks so gross!

What can I do? What should I use?