I understand your confusion regards white tongue. Here is my opinion and advice for what it's worth.
I think white tongue is almost always Candida. I think stool analysis, while useful, regularly produce negative results for Candida even when people have an invasive fungal Candida GI infection. This mostly happens when patients are on a low carb diet, which reduces yeast overgrowth, leading to a negative stool result. It doesn't mean you don't have fungal Candida infection though.
You're not convinced? Try some antifungal lozenges or other oral thrush treatment and see if your white tongue responds. Fungilin lozenges (UK, AUS) are good, but they aren't available in USA. You could also try lactoferrin powder (empty a Life Extension capsule).
I used to be unsure about whether my white tongue was Candida, but it responded to Fungilin lozenges (amphotericin) and Biotene mouthwash (contains lactoferrin). It didn't responds to nystatin, coconut oil, garlic and some other antifungals. This was very handy to know because it gave me a good idea of what AFs would work on my intestinal candidiasis. This proved to be true because I get big bowel yeast die-off from amphotericin and lactoferrin, but not at all from nystatin.
Look at your white tongue as a blessing. It enables you to test which antifungals kill your pathogenic strain of Candida.