Pilot stays on on its own. Means thermocouple is okay, pilot gets fuel. Have you checked pilot tip for carbon?
When burner kicks on does pilot size decrease?
Are there any air adjustments? Flame picture shows a little too much orange.
Can you vary temperature settings and see if anything changes?
Have you gently tapped all control units?
Are there any kinks in fuel supply lines to the water heater.
Is the thermocouple seeing too much of the main burner flame?
Does this occur just as you are getting sudds up and relaxed in the shower??
Egon