Sounds like you are about a level past me. I haven't yet had any lessons on crossovers, but I sometimes try forward ones with varying success.
Skating outside doesn't help me either much since the ice is drastically different from day to day. The ice may have large deep cracks, might have frozen bumps protruding where snow stuck, might be mushy if too warm, the lake actually tilts slightly where wind froze it thicker on one side than the other, Zambonis are rarely used (don't exist on lakes), the ice might be marshmallowy if it froze too quick, etc. The last one is hardest to explain. But it just is really sticky and slow. I can tell why the Beijing speed skaters complained about the ice being slow with the rink's new refrigerant method. The rink used captured CO2 instead of hydrofluorocarbons, but the CO2 cools the ice much faster making the ice a completely different feeling and really hard to skate quickly. You basically skate through a crust of ice instead of skating on top of ice.
Of all activities that I've tried learning, the most important thing by far is to practice on your own. In my classes, the people who practice learned twice if not three times as fast as those who only came to classes. At least with the two skating rinks that I took lessons at, there were another set of lessons right afterwards, so we'd have to leave, eat lunch, and come back for a paid public skate.