British English has many words that make eagerness sound slightly suspect.
Keen: Eager. “He’s very keen” can be a gentle warning.
Trying too hard: Making an effort that others can see, and that therefore fails.
A bit much: Too enthusiastic, too friendly or too loud for the situation.
Full-on: Intense in a way that tires other people.
Smarmy: Friendly or flattering in an insincere way.
Salesy: Sounding like someone who is trying to sell something.