Pretending to be swamped so you seem important just trains people to stop asking you.

Performing busyness buys a brief impression of importance and teaches people to route around you. The requests stop arriving, which feels like relief for a month and then feels like being left out. Availability is most of how people decide who to involve in anything. The reputation that builds is not the one you were going for.

Where you are genuinely overloaded, say so with a real timeframe instead of an atmosphere of stress. Where you are not, take the thing. Being useful is a slower route to standing and it holds up much better.