This, if the seat post wobbles in the frame or drops rapidly down without the clamp present then you can be pretty sure the post is too small for the frame.Did you check wobble?
27.0 posts are hard to find, you could have a goldmine thereI have personally measured seat posts at 0.2mm under-size. I.e. 27.2mm post was actually 27.0mm. A pair of cheap digital vernier calipers would confirm.
I've had a couple of frames with slippy posts in the past, they have always had the 'spring-clamp' seat collar and once I changed them for bolt style collar the problem went away.I had a bolt in the previous clamp but it broke apart as I tightened the allen bolt. Just noticed it today so I changed back to what the bike came with.