"Full sun" refers to the plants adaptation to light levels from shade. The shade is from things like trees or buildings blocking the sun not weather.

Plant species have many different requirements for their seeds to germinate. Some species need cold temperatures (freezing) to trigger germination. Some need warm weather within a certain range. So cool wet conditions could be ideal seeding conditions depending on the species. Look up the germination conditions for each species you want to plant.

Make sure you know your zone and the flowers' zone. Planting can vary based on that too (although, you probably already got that part handled!)

I think the seeds are likely to call for being sown with something like ⅛"-½" of dirt above them so really would be in the dark until they sprout. (some seeds - like veggies in a garden - you poke a hole in the dirt and put the seed down in the hole, and brush dirt back into the hole; some seeds you scatter on top of the dirt and maybe lightly pat into the loose dirt or maybe scatter a little bit more dirt on top)

As has been mentioned, "full sun" or "full shade" would refer to the sunlight being blocked by something large like a tree or a building. If you have a yard (or neighborhood park) that is hit by sunlight most (usually 6+ hours) of the day, with no trees or buildings or walls casting shadows, that should work well for these seeds.

Also, with the rain you don't have to water them as much!