What are Dewdrop Crystals used for?

Crystals alone won't get an Aniimo past a cap: it has to reach the level first. An Aniimo may need levelling before its resonance can be trained, so the order is levels first, then the resonance upgrade that carries it past the cap.

The crystals also sit alongside a second limit on your account. The level cap itself is raised through promotion, such as completing four core courses in an Elite Training stage to attempt Title Promotion. Resonance on a single Aniimo is a separate step from that. If your team has stopped gaining levels, check both the cap and whether the Aniimo is waiting on a resonance upgrade.

Falling behind costs you. Enemies of a much higher level gain damage multipliers and resistances, so you deal far less damage and take more. Main story progress can also stall on a required level or a level cap that needs promotion. An Aniimo held at a cap because you have no Dewdrop Crystals is one of the ways you end up in that position. Keep a stock for the Aniimo you actually fight with rather than spreading them across the whole roster.