Adam, yikes
. Two things to figure out on the carb and one regarding the tank position as per Al.
You mention the adj jet. I have had some poor experiences with these and I don't like them because they can leak. We will get to the adjusting procedure later once the leak is fixed. Is the leak coming from the adj jet?
RE the floats and the needle & seat. If it is leaking from the inside and not the adj jet removal will be necessary. The "bowl assembly" can leak in 2 ways. One way is a hole in the float from tweeking, this can be checked by shaking the float of feeling the fuel in it rolling around. The float can be drained and soldered shut again although a new float is first choice.
The second is the needle & seat (N/S). Take a good look at the needle, is it a soft tipped or brass? Then look into the seat at the bottom with a magnifying glass. The seat should look clean and free of nicks. The seat can be dressed if it is compromised and we'll cover that once we know what is causing the leak. Also the seat must be firmly "seated" to seal the threads usually with a fiber washer.
I have always suggested a pressure gage in front of the carb. However the one thing I do is mount a valve on the exit side of the gage for shutting the fuel off to let the engine run out of fuel. No leaks and it keeps the carb cleaner
.
Your on the right track.
Dave Neptune