Welcome to the forums. I agree with Der Alta, you should be able to win this series with an Integra. It sounds like your tires are too soft. Are you using racing tires? What kind?
Anyway, have you also put in a racing trans or other drivetrain upgrades and know how to adjust the gear ratios? That makes a huge difference, in my opinion. I don't know if you're doing this, but I think a lot of people go for the HP-boosting upgrades and neglect those to the drivetrain.
Anyway, here's an easy way to set up the gears, if you don't know. Before you start a race, take the car out and qualify, paying attention to what gear the car's in at the longest stright and how close to the redline. A general rule is to have the car at or around the redline (depending on where the car's peak power is) in top gear shortly before you begin breaking after the longest straight. Make sense?
You can use the auto setting for this. If you're not hitting the limit on a track, shorten the gears (moving the auto setting to the left), and so on. So, for the Test Course, you should have that setting pretty far to the right (wide). At Monaco/Cote D'Azure, it's going to be pretty far to the left.
Hope this helps/