Get a cheap 12 volt test light. Less than $10 and more forgiving when chasing tail light.
You need to make a good ground somewhere. Use a scrap piece of wire if necessary. Once you have the alligator clip connected to a good ground you can probe the socket first since you changed the bulb. You could easily have purchased the wrong bulb or not have it inserted properly. A simple test is to remove the same bulb from the left light and put it in the right side. Test your old bulb in the left side as well.
If you still have the issue then you should really start at the trailer plug side of the tow vehicle. Could be a blown fuse in the towing vehicle.
You need to be careful to not introduce a new problem while trying to diagnose this one. Take pictures as you go and print some out. Document what you did.