This is my 3rd time going to the dentist during this pregnancy and having to get local anesthetic will all of that harm my baby? I'm 35 weeks

No, The small amount of local anesthetic at the dentist is much less than you would receive in an epidural.
Drugs in pregnancy. It is typically unethical to study the effects of medications in pregnant women and information concerning safety comes primarily from case reports, (class 'b' safety is good). Typically, adverse effects on a fetus occur in the first trimester of development, your in the third trimester.
It is recommended. Pregnancy is not a reason to defer emergency dental care or treatment. Dental local anesthesia and treatment are safe and essential during pregnancy. Delaying oral care can have serious consequences. Good luck!
Ask your OB. There is typically minimal problems with local, especially when given minimally and even without epinephrine. Unless the condition is infective, painful or could lead to infection you usually avoid treatments beside preventative during pregnancy. The 1st tri is usually avoided as well as late 3rd, but correcting a problem should be done even if pregnant. Ask your md for specific concerns prior.
Pregnancy treatment. Local anesthesia will not harm your baby. Some dentists and obstetricians may prefer that you get local anesthesia without Epinephrine in it. You should always check with your own obstetrician and have your dentist discuss any precautions with both you and your ob\gyn. That should put your mind at ease as well.