From my personal experience, I usually feel better when I work out if I only have symptoms above the neck. If I feel worse during or after the exercise session, I back off. I also monitor my pulse first thing in the morning. If it’s 10 to 15 beats per minute above my baseline, I reduce or skip my workout. If the illness is below the neck, I take rest days until the illness resolves.
One of the concerns with working out when you are systemically ill with a virus, especially with muscle aches, is that the virus causing the symptoms may attack the heart and leave you with heart muscle scarring or a myocarditis. I do not know that this concern has been proven, but I worry about it. We do know that the inflammation caused by a virus attack extends to the blood vessels, including the arteries around the body. So pushing your training or competition through a systemic viral illness probably is not safe for your cardiovascular system.
When a cold lasts three weeks at this time of the year, I also think about seasonal allergies to pollens. If you do not improve you might see your personal physician to see if you should be on an antihistamine or some other allergy medication. Also three weeks is the time when antibiotics may be considered for sinus infections, so a visit with your physician may help determine if you would benefit from antibiotic therapy.