What does "lead by example" mean?

What does "out" here refer to?

"Out" here refers to the situation. This sentence could also be worded as, "But don't you take the easy way out of this situation."

Does “gross” have stronger meaning than “dirty”?

Yes. "Gross" means disgusting. You can call a person's actions gross and you can say something is gross. This has an extremely negative connotation because someone is disgusted by something or someone. Ex: Eww! He is picking his nose. That's so gross! Ex: She smells so gross! When was the last time she bathed? "Dirty" means that something is unclean. This word doesn't have as strong of a negative connotation as "gross" does. Ex: His room is very dirty. Ex: Your floor looks dirty. You should clean it.

What does "dime piece" mean?

"Dime piece" is slang used to describe a very attractive woman. A dime is ten cents, or a 10 on the ten-point scale. Ex: Megan Fox is a dime piece. Ex: She's a dime piece, I don't stand a chance.

What does she mean by "see things wrap up"? How is it related to drier weather?

She uses the phrase "wrap up" in this context to refer to the weather clearing up a little and becoming less foggy, rainy, or cloudy and instead, a little more dry. However, "wrap up" is not the best phrase to use in this context and it would probably make more sense to say "clear up".

What does she mean by "sectional"?

A "sectional" is a sofa made into "sections". It can be pulled apart and used as separate chairs or sofas. Ex: I've been thinking about separating the sectional so that there's more room to walk around the living room. Ex: I like your sectional! It fits nicely together in the corner.