It makes sense to feel rough in the morning. When we are awake and mobile, our body is moving secretions upward constantly, many time we can remove those secretions. If our nose is running, we blow it, or it ends up down our throat (yeah, I know, not pleasant) and into our GI. When we are sleeping, if we have any sort of congestion, it is likely to make its way into our airways instead, and ultimately, our lungs. Further, we are not mobile, secretions find a nice cozy home in our lungs. The combination leads to increased sputum production in the morning, some tightness, and for some, probably general fatigue.
Other than complying with medications, I don't know if there is a lot we can do to really circumvent the cycle of waking up feeling cruddy. I always wake up and start hacking for about the first half hour or so. I also have a hard time opening my eyes because I am so fatigued. I guess I just get used to it though.