Xanthan gum is a stabilizer / thickener, not a preservative.
Are you applying any type of heating or cooking? If not, that's your first step. You need to apply some type of process to not only eliminate spoilage bacteria (mold), but potential pathogens from the eggs. You mentioned fresh eggs so I'm assuming raw eggs.
Here's a reference for egg pasteurization, including liquid eggs. https://www.aeb.org/...tion_Manual.pdf There's no set temperature and time limits as it is dependent on the solids and the viscosity of the eggs. Now, if you are going to heat / cook everything with the ingredients that will change the heating temperature and time required.
Is the mold spoilage all on the surface? Or are you finding it inside the product? If surface mold I would point to poor air quality in mixing, and or packing the product. If the mold is inside the product then your heating / cooking step is not sufficient, or you are using an ingredient that has high mold levels, or both.
Welcome to food science! It can be very challenging...but rewarding once you start to make progress.