I'm sick and I'm keeping it to myself.
What is making you think you have to keep sickness to yourself?
Because it is gross and I'm embarrassed
There is no need to be embarrassed because everyone gets sick. You need to tell others when you are sick. They may like to help you.
Eh, I only have a cold. This is nothing.
You should tell people when you are sick because some may need to avoid you so that they don't also get sick and others may be able to help you.