Partial:

See me once, I likes riddles so don't be a buzzkill
I think is BEE, referring to OP, Cashbee, as they like riddles, and also buzz like a bee.

See me twice, tell everyone because I'm really ill
However, I can't see how BEEBEE, BEBE, or BB relates to being ill...

Are you a:

BEE

See me once, I like riddles so don't be a buzzkill

BEE, referring to OP, Cashbee, as they like riddles, and also buzz like a bee.

See me twice, tell everyone because I'm really ill

BB which stands for Bardet-Biedl syndrome, an illness.