10-3 is the right size for a normal domestic clothes dryer, and 30 amp is the right breaker. You say that you already swapped out the breaker, so that answers my next question of are the lugs tight on the breaker. I'd call slightly warm to warm quite normal. I've been drying laundry all day, and my dryer breaker is quite warm. Breakers will trip if they get overly warm. The fact that yours has not tripped means that it is operating within designed tolerances.
"Nothing is foolproof to a sufficiently talented fool"