You really need to go to the doctor to make sure this infection doesn't affect the baby. If a temperature becomes too high or infection becomes too bad, it can put the baby in danger. I've been through 2 high risk pregnancies so trust me when I say it is better to be safe than sorry.
Also, considering that you are already 37 weeks, even if your doctors believe delivery now is necessary, your baby will be safe and his/her lungs should be fully developed.
Tonsils and adenoid infections are not something to play around with. With your health in danger, you can literally be putting your child's life in danger. It's better for you both to be checked out thoroughly and by a specialist (ob/gyn and/or - most recommended... a specialist who has a history in high risk pregnancies where a mother is sick during pregnancy) to ensure your life and the baby's life and both your futures are safe and secure. This is NOT the time to be taking the low road with regards to health. It's not just your health you are putting on the line now. It's yours and that bundle you are carrying inside you.
Just my opinion from 24 years of experience....