What is the etymology of the shift dress?
The way OED is laid out gives some idea.
IV. Change, substitution, succession.
†7. Change or substitution of one thing for another of the same kind. Obs.
†8. a. A plurality of things of the same kind that are or may be used successively. Obs.
†b. A set or suit (of sails, scenes). Obs.
9. a. Change (of clothing); concr. one of several suits of clothing, or of several garments of the same kind belonging to one person. Obs. exc. dial.
10. a. A body-garment of linen, cotton, or the like; in early use applied indifferently to men's and women's underclothing; subsequently, a woman's ‘smock’ or chemise. Now chiefly N. Amer.
Thus it's possible to see how the original meaning of underwear came to be called a shift — the garment is changed regularly.
OED has a note:
In the 17th c. smock began to be displaced by shift as a more ‘delicate’ expression; in the 19th c. the latter, from the same motive, gave place to chemise.
It is certainly the case that 10.b, "A straight loose dress," only dates from 1957. But the straight, simple shape of that shift is not dissimilar to the simple shift of the fifteenth century. It's not so much a direct descendant etymologically, more named by analogy.
My go-to source for such questions is the Online Etymological Dictionary, aka Etymonline.
Here's what it has to say about shift in the sense of shift dress¹.
"body garment, underclothing," 1590s, originally used alike of men's and women's pieces, probably from shift (n.1), which was commonly used in reference to a change of clothes. In 17c., it began to be used as a euphemism for smock, and was itself displaced, for similar reasons of delicacy, in 19c. by chemise.
That probably makes me suspicious², but it's the best we've got. Etymonline is saying "shift dress" comes from the earlier sense of shift (ibid), to move.
TL;DR: Etymonline speculates that shift dress comes from a metaphor of moving (shifting) into a new set of clothes, as in "changing underwear".
¹ Your OED attestation from 1950s specifically describes a slip as "loose", but this appears to be a newer sense of the word, which may explain the discrepancy between that citation and Etymonline's 1590s. For example, smock, an even older word, did not develop a sense of "loose" until the early 20th C (see below).
² The more interesting thing I took away from these etymologies is that shift was originally a euphemism, displacing smock³, which had started to feel like an indelicately direct way of talking about underwear.
In turn, as the wheels of the euphemism treadmill spun through the centuries, shift was itself displaced by chemise. Nothing like speaking French to mask indelicate details!
³ Now smock has a much more satisfying etymology. It appears to come from older words meaning creep, slip, or press close, as in "a tight garment one creeps or slips into".