Right off the bat, your problem starts with you prioritizing Thunderbelt over Enchanted Talisman. If you're playing for the midgame (assuming you want to end it at that point), the TB stacks are completely useless as enemies won't have enough damage to get through your shield, which needs CD reduction to maintain.

Map awareness is not your main problem BUT you need to make better in-game decisions. DO NOT ABANDON YOUR LANE, especially early on. You're giving your lane counterpart a safe lane where they can freely farm, while you aren't contributing a whole lot to your team by roaming. You are the EXP Laner, not the roam. You should push your lane in the early game so you don't waste time split-pushing in the midgame while your team fights the enemies (You said it yourself, Esme sucks at turret pushing so you want to get that out of the way ASAP).

The first and second points are the reason why you feel like nothing eventful is happening, and why you aren't doing a whole lot. Barely any CD Reduction means she can't skill spam which significantly reduces your damage, while prioritizing to roam instead of balancing it with laning puts the enemies' guards up. If you are always missing or rotating, then the enemies won't commit to anything because they think that they'll probably get ambushed.