Answers here all over the board.
The pilot (not air) adjustment screw on the bottom will not affect any part of the throttle range other than 0 to @ 5 %. Beyond that throttle position, the slide cut, needle and needle jet come into play (3 to 85 %). Then the main jet is the only metering system in play (70 to 100%).
Valve adjustment problems are not limited to any operating range...usually hard or no-start issues OR excessive valve noise.
You use a pressure tester to check the needle and seat in the carb...hook hose to where the fuel line attaches and pressurize to 3 or 4 psi. This is the most critical thing in carb rebuild. Without a float system that maintains the fuel level in the bowl correctly, you're wasting your time with any other metering circuit. The seat is cast into the carb body on this unit. It has to be chemically cleaned and polished before you can expect a good needle to seal.
That all said, I think I'd focus on the intake/carb system. Be sure the air filter is correctly mounted and serviced. If using a K&N filter, replace it with a correctly serviced oiled foam unit. make certain that the air box is intact and the top is secured. 300's use a CV carb that has to have a certain amount of vacuum created on the back side of the carb in order for the slide to lift correctly. Also, the 300's have a small pump attached to the bowl. Check the diaphragm. When bad they run rich. Although I use carb kits often, I use no more of the parts in them than needed. I find OEM parts to be more precise (but expensive). I rarely use the jet needle or needle jet in the kits as they can be FAR off from the original and give hard-to-diagnose mid-throttle problems.
There could be other issues, but given the description of your problem, this is where I'd start. I'm curious to know the outcome. Keep us informed.