you don't need to sign-out on any computer unless and until you start the sign-in on a 3rd computer. (and you will be prompted on that 3rd computer to select one fo the other two where you are still signed-in.)
and you will lose no work IF you've saved any in-progress adobe files. ie, if you open photoshop and make some edits on something and fail to save, yes - you will lose those edits.
otoh, if you save that file, you will lose nothing.
p.s. i don't think the adobe website, and forums in particular, are easy to navigate, so don't spend a lot of time searching that forum list. do your best and we'll move the post if it helps you get responses.