Is the prayer of one who doubts her bowing acceptable, and is it counted, knowing that she experiences a lot of obsessive whispering?

Your prayer is valid, and pay no attention to doubts, because you suffer from whispers (waswasah), and hardship brings ease. Most of the doubts of one who suffers from whispers are mere illusions, so they should be disregarded to cut off the whispers of Satan. The general rule is that doubt in an act of worship necessitates returning to what one doubts having omitted. However, one who suffers from whispers should not pay attention to them, as a means of alleviation and to قطع (cut off) the whispers. Rather, they should proceed with what they predominantly feel in their heart. Among the beneficial ways to treat waswasah is to be present in heart, attentive, and ponder what you are doing. If you are confident in your action, you will not give in to the whispers.