Hi, can you say is this a sulphur (rotten egg) smell or maybe barnyard smell? You are saying that there is a smell even if you don't pass gas but does gas also smells?
About sulphur gas:
http://www.medhelp.org/user_journals/show/16645?personal_page_id=801
Sulphur gas may be from small intestinal bacterial overgrowth (SIBO) - what can be confirmed by breath test with lactulose done by gastroenterologist. Smell iteself should go away with low-sulphur diet (avoiding red meat, eggs, certain sauces, etc).
To avoid constipation:
1. drink enough fluids
2. eat foods with enough fibers: bread, green vegetables, cereals, avoid sugary food
3. be physically active, walk or so
Eliminating sugars is really a good idea. Sugars "feed" intestinal bacteria that produce gas. Elimination sugars might also help you in constipation, so try low sugar diet:
http://www.medhelp.org/user_journals/show/118248
In low-sugar diet you eat foods low in glucose, fructose, lactose and sucrose. This is the first thing you can try.
If this won't help, find a list of foods high in sulphur, and try low-sulphur diet.