Hiya, sorry to hear about the issue. It seems like it is a little low. Could be your coil. Checking the 350, looks to be similar to my 450 for charging system. Magnetic on the rotor, then coil outside with a series of windings. Possibly one of your windings could be poor or a bad connection.
I do not think anything is burned out because you still get some charge, just not enough. As for the battery draining, idle it does drain slowly since until it gets up a bit in rpm, it does not meet the charging demands. At this time, I do not have handy what the resistance for your coil should be, but start with using some electrical cleaner and check your connections and wires for the alternator system just to be sure it isn't just a loose or corroded connection.