The seed availability is indeed seasonal, so right after harvest there tends to be a fair amount of seeds around, but most tend to be gone after planting.
Logically, seeds that you eat (peas, grain, etc.) should be kept for internal use until harvest, but that stock shouldn't be available for sale, so you won't see it.
I don't think seeds disappear completely, but they definitely become rarer "while in use".
Turnips grow fast enough that you can get two harvests if the winter doesn't drag on, which means you ought to be able to get some seeds after the first harvest and plant it to be able to get a harvest of your own in the same year.