How far out from surgery are you?
Try some electrolytes. I find it helps. Whenever I'm feeling parched or dehydrated I add some electrolyte powder to my water, as it seems to help the cells and tissues take in the hydration better. Also, there are products by Biotene that help a lot with dry mouth, like mouthwash that doesn't have the more drying ingredients in it most brands do.
I have dry mouth due to autoimmune diseases, but the first few months out from surgery it was way more severe... I think the surgery impacts it somehow. I'm 5 months out and it is back to its normal level now, but I also am careful to use products that don't contain things that make it worse.
Find a good lip balm that has ceramides in it, those help the skin barrier repair itself. There are a few good ones out there. If you need a suggestion I can go hunt them down, just let me know. The one I'm using right now is some Korean brand my sister got me for Christmas and I love it!