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You said you are not eating too much, but are you eating enough each day? How many calories are you consuming throughout the day? If you are not well-nourished, this could also cause dizziness (as well as the information Cary provided to you above).

Goat and sheep milk are still dairy products, so I'm not sure if your doctor asked you to cut out all dairy products, or just eliminate cow's milk? Soy and almond milk are non-dairy substitutes.

A balanced diet can include a diet without animal products (dairy, meat), as many vegetarians eat a balanced diet and are healthy. There are different types of bread, also, some that are sprouted or some that are gluten-free. As long as you are eating a balanced diet that includes healthy, lean proteins (beans and legumes are good choices), plenty of fruits, vegetables and water...plus enough calories to get you through the day...your diet may not be causing the dizzy feeling and it could be related to another condition (as Cary said, you will need to talk with your doctor about this).