A., don't give your child advil. Advil is for children 6 months of age and older. It can cause serious problems with the kidneys if given in children younger than 6 months. Additionally, advil only treats things like fevers and inflammation, and not colds...
Why don't you have insurance yet? In many states, you can get insurance for your child even if you don't have a job or if your job offers no benefits.
I would recommend that you see a doctor right away. Online resources like MamaSource are good places to start but they do not substitute for doctors. We don't know what your child looks like. We can't do physical exams. As you say, don't go to the ER if you feel your child is not that sick. You'll be stuck with a bill that you may not be able to pay. Find a local pediatrician, make an appointment, state that you don't have insurance and ask can you pay for a visit out of pocket. An honest pediatrician will charge you no more than $45-75 for what's called a sick visit. Then, make sure that you don't find yourself in this position again -- get your insurance situation straightened out ASAP.
On the other hand, if your child is really sick, you have no choice but to go to the emergency room, but prepared to pay a bill. While you are there, state that you have no insurance (or you need help to get it ready to use right away) and usually a social worker should be there to help you with that. If you really have insurance but paperwork and stuff like that is pending, they may be able to use your insurance right away and you won't have a big bill.