I am very particular about washing my hands if I touch something meaty/milky, before I would touch something the opposite or parev. That is OK for an adult, but I can’t run after my kids every few minutes telling them to wash their hands after every little piece of food they touch, and telling them not to touch other things. This theoretically can cause problems, but in practice, what should one do?

There is no Halachik restriction touching milk products then meat or vice versa. However, one should be careful not to mix these two substances; therefore if after touching meat, for instance, your hands are greasy – you must wash them before you touch milk products lest a mixture of the two will reach your mouth.