is there a way to move my color swatches I gathered while working in Photoshop to Illustrator (both versions are CC)?
You can save swatches to Adobe CC library (See this link for a full introduction to using the CC Library). This library can save various assets but one of them being swatches which you can use on various Adobe programs. The swatches will sync whenever you make a change to the swatch.
There are many ways to add swatches to a library.
1) Add assets from the page by clicking the "Library from document" button
2) Add Foreground Color button
3) Drag a swatch from the swatches panel into the library window
4) Make or download swatches from https://color.adobe.com/