My glands are swollen and I am in horrendous pain; no painkillers are working. I went to the dentist today and got amoxicillin. Could ER give me pain relief?

Depends. Sounds as if the dentist was trying to control infection prior to root canal or extraction treatment. Pain killers may cover over resolution of your infection problem. See an endodontist (root canal specialist) or an oral surgeon asap. Until you do, keep taking the antibiotic.
Pain relievers yes. The Amoxicillin will probably need 48 hours to quell the pain from the infection, so by all means i'd consider going back to your dentist ( or doctor) and getting a prescription for pain relievers! you will only need about 12 or so to get you through until the Amoxicillin takes full effect! i hope you feel better soon!
Get back in immediat. I sounds like a "hot endo situation" to me. She probably needs the tooth opened up. If it is something that needs general anesthesia and deeper procedures, an oral surgeon is the guy to see.