It seems now that anytime I "shop" online, i.e., look at anything in a shopping site such as Amazon, ads for that thing I looked at show up everywhere. For instance, I happened to click on ladybugs and now I have ladybugs showing up everywhere. These things last for weeks even if I actually buy the thing! I'm sure they have something to do with cookies. Is there anyway to remove that cookie without disabling my ability to shop on Amazon?
First of all, try blocking Third-Party Cookies. Most of these retargeting ads are run by scripts that won't be able to drop their own cookies if they aren't from the same domain as the site you're visiting.
There are a few more things you can do, depending on your choice of browser. Adblock-type plugins (available for Chrome or Firefox) will often block the scripts that try to drop these cookies, and plugins like DoNotTrackMe & Ghostery try to block even more for you.
Firefox third-party cookie settings:
Chrome third-party cookie settings: