I use Premiere and occasionally Photoshop via my institution-wide licence. Last year I was able to use my school account on three different computers, but recently I had trouble launching Adobe apps on two of the said devices. I went to the IT administrator, who claimed that as a result of some new update to CC I will only be able to log in to CC on a single computer, and will have to log off to use it on another device. Is there any new policy like this in place? And if there isn't, how do I use CC on multiple devices without having to constantly log in/log off to it?
Your license can only be active on two computers at the same time and you can only be logged in to your CC account on one of those computers at a time. If you move to a different computer, you will have to log out of the CC on that one and log in to CC on the other. There is no way around this.
I'm not certain what you mean by an "institution license". However, if this is a license you hold in your own name, it's possible that you have activated it on too many machines. Creative Cloud can be activated on only two machines at a time, and you can actively use it on only one of those two machines at a time.
If you've logged in to the CC on more than two machines, you will need to find those other machines and log yourself out of them. See the below link for more information.