If I am sick and have nasal congestion, will this worsen when I fly because of the altitude?

Ears not nose. The problem you are more likely to have is with your ears. They won't be able to 'pop' to react to the pressure changes if the nose is very congested. Consult with your doctor or an ENT to see if using a decongestant might help you with the flight. Or, postpone the flight so that you aren't a risk to both yourself and others.
Yes. If your sinuses are full and swollen there is a higher risk that your eustachian tube will be swollen also. This prevents fluid from draining out of the middle ear and can cause a lot of pain when traveling. If you are really sick, you would also expose everyone on the plane to what you have because the air is continuously sent back through the cabin.