It is known that if a multiple-access arbitrarily varying channel is symmetrizable, then the capacity region has an empty interior. In this paper, we show that if a suitable constraint is placed on the channel state sequences, then the capacity region can contain certain open rectangles and thereby possess a nonempty interior. We also prove a new weak converse under a state constraint. We use our results to establish the capacity region of the two-user adder channel under state constraint 1/2.