Sounds like you have true greek oregano which is a creeper and many people(including me)prefer the flavor to common oregano. It will eventually put out upright stems with white flowers, but they won't get too tall as you noticed.

My oregano only gets tall during the warm weather. Yes ! it loves to run which is just fine if you find friends who also enjoy this herb. I believe it is hardy up to and including Zone 5. In 3 years my oregano went from a 3 inch pot to a four foot by four foot bed. I found that when it gets tall and is not cut back you end up with a lot of dead shoots come the next spring season. They will need to be cut back or you will have a messy looking herb bed. It does not effect the rest of the plant it is only unattractive looking.