I have the same problem off and on. I've found I need to drink LOTS of water (I've cut almost everything else out) to stay hydrated, which helps. I've also been through several brands of lip balm; I like Blistex the most. There's one in a blue tube which is great, and there's one in a white tube that's like lotion for your lips, it's also great. I put vaseline on my lips every night as well. You don't need much, but it does help.
Avoid licking your lips, all that does is dry them out more. You're mom is probably right that you don't drink enough water.... so start!

For the lip balms you just need to reapply when it wears off, eventually your lips will re-hydrate and become healthy again.
Oh, and fight the urge to peel the flaking skin, it just causes more damage. You can buff your lips gently with a damp washcloth at night to get rid of the flakes, that will help. If you do that and then apply vaseline, you'll find it will help a lot.