If it's been stored for 10 yrs., it's possible it may have crud in the fuel tank, when you added fresh gas it may have stirred it up and then the carb will continually become clogged.
Drain some fuel in a clear container and inspect it, if it's polluted, you'll have to drain and clean the tank and carb again...(totally dismantle the carb)
Install a small fuel filter in the fuel line (available at bike shops or mower repair)
The throttle grip can be tightened by the screws in the metal part of the grip (where the cable comes out). A close up photo would help.
If it's just the rubber part of the grip that's slipping on the plastic tube, remove and clean any oil or grease from it then reinstall with rubber cement.
If the cable going to the carb has any play in it, there should be a cable adjuster near the throttle grip or where it enters the carb, to take up the slack.
Hang in there, you eventually get the bugs worked out and have a fun set of wheels and a neat conversation piece.