[RaggedMountainMan]

All above sounds spot on.
Does sound like float needle valve is stuck in the open position, perhaps ever since the mechanic "rebuilt the carb" and issue then was only revealed when petcock was opened for your ride.
Having someone do the work is nice and convenient only if you can trust them to do a better job than you can. Can be hard to find someone familiar with TWs though.
Disassembling and inspecting/ replacing the float's needle valve is not difficult and makes for a great learning experience. The more you understand your TW the more you may appreciate these litle machines.
Maybe I'm goofy, old fashioned or just curious about the world I live in but I feel the need to understand any system I am interacting with. Since bad things can happen far from home this understanding seems especially true for a motorcycle.

After all you don't want to be like this guy I came across middle of nowhere Nevada sitting amongst all his worldly possessions next to his blown Triumph. Not easy to find only available shade uncer the bike when you have to share it with a growing oil spill. Unfortunately he knew almost nothing about bikes, was moving from east coast to west coast and new no-one in between other than this strange guy with a Betty Boop decal on his bike who stopped to help.
View attachment 234800

His first mistake was going that way. What did he expect to do about gas, oil and breakdowns?

 

[RaggedMountainMan]

I’m trying to remember, but I don’t think the drain screw on the carb is the correct one. I’m trying to remember what my 2006 had….. my 2023 is a crappy little jis head. But the above answers are spot on-something’s up with the float needle.

Yip! Agreed. Tap, tap tap. Something stuck in the needle seat or needle stuck open. I've seen them actually cocked and stuck at an angle. May need to pull the carb and set the float level.