I don't know about Bag Balm, but I personally swear by Burt's Bees Lemon Butter Cuticle Creme. It comes in a little tin and all you do is rub your fingers on the waxy stuff and then massage it into your nails and cuticles before bedtime (do it every night whether you've been potting or not). It helps moisturize without petroleum products, helps your nails to grow and strengthen, and also works on those little cuts as well as chapped lips. Lemon has antiseptic properties, plus it smells refreshing and tastes great! It's all natural as well and easily found at many stores including CVS.
If you don't want to spend money on the Lemon Butter Cuticle Creme, go with Extra Virgin Olive Oil. Rub a dab into each nail/cuticle until your hands don't feel greasy. If you do it at night you don't have to worry about contaminating your clay, etc.
PS, clip your nails...don't bite! That can worsen your problem. But I do echo the sentiment about keeping them short.
Years ago, I had a colleague that taught in the same studio as I in a different period. I would often be in the room during my prep period getting ready for my next class. He complained about the cracking of his hands that would get really bad. I had the chance to observe his habits, and found that he would wash his hands with soap and water after nearly every thing that would get his hand dirty-often 4-5 times in a 50 minute period. He would teach 3 periods. One day I told him to only rinse his hands, except before eating and after restroom, as he was removing the protective oils from the hands. He tried doing that, and found that his hands healed up in a month. Next year no cracking. I always use a little hand cream in the studio, but don't obsess with it.