I also wear shorts over my tights in the winter (spandex is not kind once you reach your mid-40's). I feel very uncomfortable in just the spandex & the shorts are just enough to make me feel better. I'll admit, it's my own personal hang-up, but I can't help it.
Most days (I live in Oklahoma) the tights & shorts are enough, but I always run with running pants over my tights when the temperature gets below freezing. I also generally wear gloves, a thin stocking cap, and two layers (moisture wicking, of course) of shirts. I find that if I keep my hands & head warm, I feel warmer. The key is dressing in layers, so you can remove clothing if you start to overheat once you get moving.