Poa annua germinates mainly in the fall and there are gazillions of seeds lying around waiting to germinate then. So the best way to get rid of it is to use pre-emergent in early fall. Early fall is also the time to plant cool season grass, so if you also want to overseed, you can't use normal pre-emergent then. Some people use pre-emergent in the fall to take care of the poa annua and then seed in the spring to get new grass. Spring seeded grass has its own issues (can't use pre-emergent to take care of summer crabgrass, summer heat overwhelms inadequate root systems). If you have lots of shade, it's always going to be a problem to get grass to grow. Have you considered alternative groundcovers? Shade garden plants (ferns, hostas, etc.)? Mulch and shrubs?