I stopped using drugs when I became pregnant. Will my baby suffer the same withdrawal symptoms I'm suffering?

Depends. If you stopped as soon as you became pregnant then your baby will not be addicted itself. You should seek medical help if you are experiencing withdrawl symptoms. Sometimes women who are withdrawing can experience seizures, etc and this can affect the baby.
Tell your OB. Since brain and heart start forming at 3 weeks, tell your OB exactly when you quit. (s)he may order high-resolution fetal ultrasounds or fetal heart-specific imaging. While these may not tell you everything, you'll get useful information and you'll be less likely to start using again to handle worry/stress.
Drugs in pregnancy. This depends on what drugs you were using. Many drugs are harmful to fetuses, but also withdrawal from some drugs (like opiates) can be even more dangerous. All drugs you take will reach placenta, & baby will be in withdrawal if you are. If you've already stopped your drug use and are staying clean, your baby will not be in addiction/withdrawal when born. Talk to your OB about specifics, please.