Are "好饱了" and "好包了" both correct?
Actually, "好包了" does not mean "I'm full".

You may see "...打好包了..." in the Google hits. It refers to "have made something into a package.

If your friend say "这顿饭我包了". That means your friend will get the bill, and you don't pay the bill.

You will see "7天包退" on some goods's package, that means "7 days to cancel purchase for non-faulty goods".

And "他们的村庄被敌人包围了" refers to "Their village was surround by the enemy."