How do I evict my son who has been paying monthly rent for a bedroom in my house but has no written contract?

You have two different issues.  The first is how to get your son out.  Unfortunately, if he was paying rent and considered a tenant or boarder-- then you will need to follow eviction procedures.  I have included a link to an online brochure by the Georgia Department of Community Affairs which sets out the basic procedures for you to follow:  http://www.dca.ga.gov/housing/HousingDevelopment/programs/downloads/Georgia_Landlord_Tenant_Handbook.pdf

Your second issue is the going into his room.  Generally-- a landlord is not supposed to go into a unit except to repair something or to fix a dangerous situation.  It's a little fuzzy, but removing illegal drugs could be considered an emergency situation.  If you regularly entered the room with his consent to help with housecleaning (picking up towels, dumping his clean clothes on his bed), then you have an additional exception through the consensual routine or practice in your home.  Either way, it's not likely to be an issue for you.  Most druggies are not going to file a criminal or civil suit against someone for disposing of illegally controlled substances they were not supposed to have anyway.  Mainly, because they could be getting themselves into more trouble by self-reporting their own illegal activities.

As a going forward, it will take some tough love, but you need to keep him out of your house until he is completely clean.  Get him into a clinic or something.  You did the right thing about getting rid of the drugs.  If the feds or some other narcotic task for knew that drugs were going through your home, then your property could potentially be subject to seizure (meaning you would lose your house).  I know your son is important to you, but he's obviously putting his bad habits above your necessities of life by putting you in this situation. 

Assuming that your son is 18 years or older and an adult he is under the laws of all states in this country on a month-to-month lease with you regardless of a written agreement to that effect. To evict him you serve upon him a thirty (30) day notice of the termination of his lease.

If he does not vacate the premises within the next thirty (30) days, your legal recourse is to file an eviction action in your county court house. If matters come to that, you should consult with a landlord tenant attorney for assistance.