The 3-month rule is solid but it depends on use - abrasive whitening or charcoal toothpaste and heavy pressure will make bristles splay faster. To keep costs down & try reputable compatible heads in multipacks and rotate two heads so each can dry fully between uses; rinse the base after each brush and avoid hot water or dishwashers that deform bristles. If you’ve been sick, replace the head early since worn bristles don’t clean as effectively.
Had a similar issue with my ceramic straightener last summer. It would turn off after ten minutes or so and no warning. Annoying as hell. Cleaned the plates thoroughly, thinking maybe residue was causing it to overheat and trigger the auto-shutoff.
That didn't work, so I tested the outlet with another appliance. Turned out the outlet was faulty, not providing steady power. Switched to a different one and problem solved. Simple fix. If that's not it for you, maybe the heating element is wearing out from all that use on thick hair.