All good advice... (see your doctor!)
It's important to define what you mean by dizziness. Is it an experience of the room seeming to swirl, even with you eyes closed, or is it more of a feeling of instability, again, even with your eyes closed. A test: While experiencing the sensation you describe, have a friend look at your eyes with your head still. The observer may well describe your eyes as making tiny but noticeable, rythmic "jerking" motions from side to side.
Additionally, are you experiencing sinus pain or stuffiness (or discharge of greenish mucous)?
My own experience as well as sharing anecdotes with others seems to indicate that most "dizziness" originates with the inner ear and the circular canal complex which does, normally, provide for one's stability. When one experiences "dizziness" and finds that closing their eyes alleviates the symptom, what's acutally happening is eliminating a discrepancy between two systems for determining ones "place in space". You've caused the circular canal to disregard any motion sensations that are not accompanied by agreement though eyesight and may result in cessation of the dizziness.
Try this; while standing, lift your chin ( do this only near a chair or something to quickly sit on) as though you are looking at the ceiling at about a 45 degree angle. While doing this, swiftly move your head from side to side and stop with your head in a normal, straight ahead position. Dizziness? Severe or mild? Does it go away or abate with closing of your eyes?
At any rate these will be important indicators to relate to your doctor and will probably result in being diagnosed as having an inner ear infection.
Such infection may or may not have any discharge, pain or other visibile indicators. Often, mild sedative medications are prescribed for a duration as well as antibiotics to eliminate the infection.
Be