Can other fungus invade the mouth and lips other than albicans that can be contagious and passed on? peeling in sheets and white tongue

Of course, rate of..: growth compared to other organisms in the location/conditions present is the principal determinant, Not exposure. While exposure is commonly talked about, local status & immune function are the overwhelming dominant issues. Fungal infections become more common when using broad spectrum antibiotics; temporary suppression of bacterial growth leaves the territory wide open to fungus or whatever else.

Fungus in the mouth: Candida has many species, with Candida albicans being the most common. Indeed, many fungi can be found in the mouth. The question is 'what' caused your yeast overgrowth, as those with 'normal' immune systems won't be susceptible. So I wouldn't worry about passing on any fungus - the body can routinely handle that, however I wonder 'what' underlying problem there may be that u need to get fixed.