Thank you for the additional information.
There are a couple of things to look for. First, check to make sure the air filter is installed correctly and sealing properly. No air filter or improperly installed/not sealing can cause the unit to run extremely lean and exhibit the symptoms you described. If you have the stock air filter and it is installed correctly, with the lid on the air box (essentially EVERYTHING is in stock configuration in the intake tract) the next thing I would check would be the size of the main jet. Compare the size # ***** the MJ of the new carb to the # ***** the MJ of the old carb. If it is a different size, put the old MJ in the new carb and give it a go. Typically when gas goes bad and plugs up the carburetor, it does not affect the main jet, because the orifice is so big relative to the size of the pilot (or idle) jet. Also, the symptoms of bad gas/plugged up jet in the carb is the unit will start and run on choke but once it is warmed up it will not want to idle off choke. If you were able to keep it running, it would run normally from mid range up, it would just want to stall when you tried to allow the engine RPM to return to idle.
The reason I was asking about if the carb was obtained straight from Eton or an Eton dealer is because many different bikes or manufacturers will use the same carb body, but have them jetted entirely differently to suit different applications, models, or engine sizes. If you do happen to find a difference in the MJ size, I would look at the other components to ensure they are what it was supposed to have installed as well.
Another thought that comes to mind is if the float level is too low. This would lessen the amount of fuel retained in the float bowl while the engine is running, and can cause lean symptoms as well. Probably your best reference on this is compare the height of the floats from the old carb to the new carb. Take the bowls off each carb, and with the float hanging from the float pin and the float pin in alignment with your line of sight, start tipping the carb (or rotating it) back in the direction of the float pivot pin. This will allow the floats to move up into the carb body. Watch them carefully and observe the angle the carb is as when the floats stop moving into the body. Compare the angle between the new carb and old carb. They should be very very close, if not identical. If you have to tilt the old carb farther back than the new carb, the float is definitely too low and will need to be adjusted so it matches the old float height measurement. You do not want to just flip them upside down because the weight of the float can compress the spring loaded contact on the float needle and give you a false reading. Always check the float height by tipping the carb backwards, as viewed from the side of the carb. Normally a tech would use a float height gauge to measure it, but since most people don't have one of those, and you do have another carb that ran fine for awhile, you can just compare A to B.
If you can't make any headway with the new carb, it certainly isn't going to hurt to rebuild the old one. I would suggest soaking the old carb body in Pine Sol until it looks nice and clean. Pine Sol (household cleaner stuff) works amazingly well to clean carbs and is usually easier to come by than legitimate carb cleaner/soak. Just remember it is usually the small jet that plugs up and creates problems, not the main jet.
On your new carb, you should double check the needle clip setting as well. Sometimes that gets set too lean and can produce similar symptoms to what you are experiencing.
If you have any more questions, let me know.