hello i also suffer from dry mouth/chappy lips. My doctor worked out it was also because I drank too much water. I now add a litre of coconut water to get electrolytes in - another trick was OJ + Sparkling water + a pinch of salt.
I use Biotene toothpaste + Mouth wash. It helps.
Re lips - please take this the right way but it might be herpes virus - as i understand (NOT A DR) the herpes virus is responsible for cold sores and well the herpes virus activates when your immune system is suppressed. I never had a visble cold sore in my life but I carry Herpes simplex (as does like 80% of population apparently)- responsible for cold sores and can present in mouth as mouth ulcers - that happened to me. Try speaking to the oncologist or nurse and see if it might be herpes related as that might change what medication they can provide?