You can just add to existing soil, and till up. You should do this now, there is no benefit to waiting. You will still want to add fertilizer when you plant your corn. It is a very heavy feeder. Sunflowers will benefit from some nitrogen.

add it now and till it under it is good to let it rest before planting and you wont loose anything from doing it earlyer.

how do I add composted manure to garden soil? I heard I should place it under the soil, then add topsoil ontop, or should I just mix it in? I'm growing corn

and should I add in my manure now? I'm planning to transfer my plants a month from now, or should I add it in just days before I plant them?

Bonus Question: do sunflowers need nitrogen?