I had almost the exact same experience in late Jan [edit: 2020], along with most of my coworkers. I did keep wondering if I had Covid so I got tested, but with a negative result. The main piece of evidence that speaks against the Covid theory is any kind of mortality deviation, which didn't occur until later half of March.

Anecdotally, over time I heard many other locales, workspaces and schools experienced a similar "weird cold." It may not have been Covid but there was certainly a very widespread, and very unusual cold in the winter of 2019-2020.