The lungs are an unlikely cause of headache or vertigo. However pulmonary hypertension that is severe enough to interfere with the pumping of blood by your heart could be related to headache and vertigo. Efforts should be made to determine the cause of your shortness of breath at rest. Independent diagnostic tests should be done to find the cause of your headache and vertigo. This is more likely to be coming from a problem in your head, ears or sinuses.
There is no connection between dizziness and you lungs unless you had serious lung disease and were not taking in enough oxygen and were faint.
I'd suggest you go to a good internist--perferably one who is an allergist who can help you figure out the cause of your headaches & try solutions. Low grade headaches & vertigo can be SO annoying.
Vertigo can be cause by a virus or infection & can take some time to clear up. Antivert is a prescription medication that helps some folks relieve vertigo symptoms (tho it didn't help me).
Starion