The old fashioned leafy lettuce are soft and floppy anyway. Next time buy a cos variety like little gem or Lorraine, if you buy another round one put the cut end in a dish of water and leave it on a windowsill.
I always wash and put lettuce in a container and cover with kitchen roll then put lid on .you will feel the kitchen roll gets damp so just put a fresh sheet on . I buy a bag of the watercress and rocket as I need it to keep all week for my 2 birds . COS lettuce keeps lovely but I have never tried the leafy round lettuce but I would think it works the same .
I’m always of the opinion if a products in a freezer/chiller in the supermarket that’s where it goes at home. Do you see lettuce, tomatoes, cucumbers mushrooms in chillers?