Jenni,
Take government issued warnings with a pinch of salt. They overestimate risks.
In the first Gulf War the British Embassy suggested that all families should leave yet were telling business men that it was perfectly safe to travel to Qatar.
We spent 17 years in Qatar in total and never ever felt at risk, even when we were told not to visit the souk! We did and were greeted like long lost friends.
I don't wish to be critical of the US Government but the US Embassy has a reputation among the older expats for over reacting. Go to Qatar and enjoy life.