Why do you think you messed up the battery?
The 350f wont charge battery much below 3500 rpm.
Why don't your speedo and tach work? The 350 is a dangerous machine not to have a tach on if you are a new rider. It spins up easily well past redline. Good way to grenade your engine. Look into fixing that.
Yes, the air screws are the flathead screws on the sides of each carb body, on the airbox side. Do you know how to adjust these?
Listen, I'm not trying to be a jerk here, but you are a new rider, and a new at wrenching, and new at understanding old bikes. There have been a lot of questions asked by you that fall under very routine maintenance. You were taken for a ride by the first mechanic you took your bike to. And since you really don't understand even the basic workings of a carburetted 4 stroke internal combustion engine, suggesting that you try this or that, without having to spell out an entire chapters worth of theory, experience, data, cross referencing, and cause-and-effect logic trees is not only difficult, but possibly harmful.
You really, REALLY, need to find someone in your area that you can both ride with, and learn to work on with, an old bike. Find ANYONE, a group of people that have a 'bike night', with people that are experienced, at least somewhat, with old bikes. DOesn't have to be old SOHC4 Hondas. Just has to be old-tech knowledge.
I wish you were nearby, because you could bring that sucker over to bike night 2 or 3 times a week, and we could teach you basic maintenance and basic theory in couple of weeks, so that you understand why you are doing something, and the steps to hunting down via experience, theory, and process of elimination, the problems you are having.
Do your best to find a group of people who love old bikes that get together in your area, bro, it'll do you a world of good.