Fiona: With sunflower seeds you dont need to refridgerate. Just take all the seeds off the head.. keep what you want to use to plant next year and then the rest can go to the birds outside. I hang my heads on my fence upside down and the birds spend weeks eating the seed. I grow giant sunflowers and every year I harvest about 40 seeds from one head and give the rest to the birds.. Once you have dried the heads out take the seeds and just keep them in a paper bag in a cool dry place.. Dont put in plastic bags because if there is a tad bit of moisture they will rot or grow mold. So paper bags are best.
Good luck!