Yes, dig up, store, and replant next year. I will do that with my dahlias as well. (I put little rock rings around them so I can remember where they are.) If you grow them in containers, you don't have to dig them up - just store them in the container.
(I have unknown daylilies growing in shade. I did not plant them and they do not seem happy where they are as they have produced just one flower stalk (scape?) so far from a row of about eight plants. You might want to try asking in the daylilies forum if nobody here has an answer.)
Daylilies do need sun to flower. Ive experimented a lot over the yrs., I would say in general you dont get any flowers in total shade, a sparse few in dappled shade, more in part sun, a lot in 1/2 day or more. Except if its a really hot all-day sun then thats no good either, they seem to fry under those conditions. Those are my observations anyway.
duckee
kim31kim
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Debbie Downer