I am currently experiencing a bug where I am moving as if I am overburdened even though I am about 100 under capacity, I can still even fast travel.
I've also dropped everything in my inventory and still move as if I am overburdened.

Make sure caps lock isn't on

Make sure you don't have a disease that is sapping your strength.

This happened to me while I was testing interface stuff today. It lasted for about 30 seconds after I loaded the save.

Load save, try to move, realize I must be in wrong mode, hit caps lock, nothing changed, tried shift to run, no change, was able to sprint (once, for about a second) but not run.

Tried going into God Mode in the console, because that removes the overburdened thing, but was still walking slow.

After all that I started moving fast again. Only happened the once but I will try to get more info if it happens again.

It was definitely not (only) the always-run setting.