When you go to a public skate, you control the time and space of your skating eg when to move, how fast, when to stop etc. So you always seem to be in control and balanced. In a game, which should be a faster pace, you are having to react to the situation eg stop, turn, accelerate etc when you are not expecting it. Your balance, skating and timing gets disrupted and until you become a better skater you will continue to struggle. Bottom line is practice, practice, practice. As you become a better skater it gets easier. Skating with a stick in hand does help a lot. Skating with kit will get you used to it but at the end of the day it's how well you skate, not the kit you may or may not be wearing.
And when you go to a public skate you need to work on your skating, not just skate in circles around the rink. Edge control, one foot balance drills, unusual skating positions, even figure skating moves (eg a 3 turn), all so you develop as broad a base as possible of skating skills. Youtube is a great friend here, type "Advanced edge control drills" into you tube, find drills you have never seen or done before then go and practice them. Or ask a coach to give you some drills, everyone I coach gets drills to work on when they go to public skates, all designed to make them a better skater.