Just got over flu a week ago. Now have sore throat/cough/headache got small kids, should I worry about being contagious?

Flu ? contagious. You may be - but its probably viral and if anything - your kids probably gave it to you. Good hygiene will help reduce the risk of spreading it - hand washing, keeping your mouth covered and avoiding sharing towels or drinking utensils will help.
Probably not... Sometimes after a cough has been present for a while, bronchospasm can result once the infection has resolved. If there is residual bronchospasm, symptoms like you describe could be present. Treatment can consist of bronchodilator and steroid inhalers until the cough completely resolves. See your doctor so treatment can be rendered.