I was having the same problem with 2.7.1 and spent all day trying to figure out how to fix it, being that I don’t like the idea of a workaround.
Amazingly, it turns out that I was able to fix the problem of the custom fields not saving by entering the data into the custom field but NOT clicking “Add Custom Field”.
Everytime I would enter the custom field data, then click “add custom field” like everyone tells you to do… it would disappear when I went to save it, and instead be saved in a draft. Without clicking that button, ALL my custom fields are saved…. and i actually have to click LESS!