It sounds like your blood sugar is reacting to something in the meal; it could be the carbs/starches (potatoes, rice, bread) rather than the veggies.
I'm not a diabetic but I find that they make me hungry too, and it helped to eliminate these from some meals, and in others to make sure to balance them with a good amount (at least 3 oz) of a protein like beans or chicken. And as noted above, a small amount of fat like olive oil, yogurt, or cheese can help too. Although some vegetables depending on how they are prepared can cause your blood sugar to spike too, especially if you eat them without protein.
Have you read at all about the glycemic index of foods? The idea (though not everyone agrees with it, it's true) is to keep the average glycemic index of your meal as low as possible, by eating carbs that are processed more slowly, for instance whole rather than refined grains. It doesn't always work as advertised for me, but in general I haven't suffered the hunger pangs I did before I started paying attention to GI values of my foods.
Here is a page from the Mayo Clinic on the
glycemic index and blood sugar
and here is the Univ of Sydney's
glycemic index page with database
but you can also just google "glycemic index
foodX" and generally get an answer that way.